Mathematical Model Of Process Research Articles

Heat Transfer Slot Matrix for Compact Rotary Heat Exchangers with Ultra-High Regeneration Ratio for Microturbines

This paper considers the problem of the influence of distancing elements in slot matrix channels on the thermohydraulic characteristics of the matrix of a rotary heat exchanger having various values of the height of the slot channels. The following tasks were carried out: mathematical modeling of thermohydraulic processes in different slot channels having constant height along them (from 0.3 mm to 0.5 mm); analysis of the influence of the slit channels' height, the number of distancing elements in rows, the distance between rows and the total number of elements on the average value of the Nusselt number and on the pressure drop; analysis of the effect of changes in speed and temperature conditions on thermal and hydraulic characteristics. The most important results are the following facts: the influence of the relative location of the distancing elements on the thermohydraulic characteristics of the flow is negligible; changes in the temperature and speed regime of the laminar flow practically does not affect the intensity of convection in the matrix channels; an increase in the height of the slot channel leads to a relatively small decrease in the average value of the Nusselt number. The significance of the results obtained is that they can be the basis for adjusting the methods of design calculations of rotary heat exchangers taking into account the decrease in thermal efficiency caused by the presence of distancing elements in the channels of their matrices. Recommendations can be made to minimize the number of distancing elements without considering their mutual location.

The Model of Ice Destruction on a Solid Surface by Spherical Striker Impact

Nowadays, there is a problem related to the difficulties of the vehicles and pedestrians’ movement in the cold season due to the formation of an ice layer on roads and sidewalks. As a variant of the problem solution, the mechanical ice destruction by impact-destructive working body with spherical strikers is applied. The use of the spherical strikers provides not only effective ice destruction but it also prevents from damage of the road and sidewalks pavement. In order to achieve the effectiveness of the ice destruction, it is necessary to use appropriate mathematical dependencies for calculation of the working body parameters. In the paper, mathematical model of the ice destruction process using spherical strikers has been conducted. Theoretical dependencies of the mathematical model have been corrected based on the experimental studies.

Catalytic reverse-flow oxidation process in reactors of various designs: Axial, side and tangential gas inlet

The study was dedicated to the mathematical modeling of the catalytic reverse-flow process in reactors of different design, e.g. in reactors with axial, side or tangential arrangement of reaction gas inlet/outlet streams. The oxidation of volatile organic compounds in the fixed adiabatic catalyst beds was considered, using toluene as an example contaminant. Three-dimensional fluid dynamics modelling performed by means of Comsol Multiphysics software showed the existence of the significant gradients of temperature, concentrations of reactants and fluid velocities along the bed radius; such gradients are neglected in the conventional one-dimensional models, usually used for simulation of reverse-flow processes. The spatial distribution of these parameters may be asymmetric in the reactors with asymmetric design. It was shown that efficiency of both toluene conversion and heat regeneration significantly decrease in reactor with side feeding of inlet mixture, finally resulting in lower thermal stability of the process with the significant (∼20%) decline of the maximum possible duration of the cycles between flow reversals compared to reactor with axial gas feeding. Vice versa, the tangential arrangement of gas inlet provides the visible (∼10%) increase of maximum cycle duration. Application of 3D modelling approach to development and optimization of catalytic processes becomes especially important in case of reverse-flow reactors where negative influence of various spatial non-uniformities may impose more significant influence on process parameters.

Development and Research of a Promising Pumpless Liquid Cooling System for Reciprocating Compressors

A new pumpless liquid cooling system for a single-stage two-cylinder reciprocating compressor has been developed from the analysis of work processes and cooling systems of reciprocating compressors, where one piston compresses and moves gas and coolant in the cooling system. The intensification of the coolant movement increases in the machine, which can reduce the temperature of the cylinder–piston group and increase the indicator efficiency and the compressor feed rate. A mathematical model of working processes in a reciprocating compressor and its cooling system has been developed on the basic fundamental laws of conservation of energy, mass and motion. A prototype was developed and tested to obtain new knowledge about the processes in the machine and confirm the assumptions made while developing the mathematical model. After a series of experiments, the influence of cooling on the working processes in a reciprocating compressor, the technical work carried out in each working process, the energy and consumption characteristics of a reciprocating compressor with a developed cooling system were established. There was a validation performed of the developed mathematical model of work processes in the machine and the assumptions made. According to the developed mathematical model of work processes, a numerical experiment was carried out, which established: the coolant flow rate decreases with an increase in discharge pressure and increases with an increase in the crankshaft speed when the discharge pressure changes from 0.4 MPa to 0.8 MPa, a decrease in the indicator isothermal efficiency due to the leakage and non-isothermal nature of the compression process, as well as a decrease in the return work in the process of reverse expansion, leading to a decrease in the indicator isothermal efficiency of more than 15%; due to the equalization of pressures in the gas cavities with an increase in discharge pressure, the coolant consumption is reduced by (15 ± 17)%; with an increase in the angular speed of the crankshaft from 800 rpm to 1500 rpm, the coolant flow rate increases by more than 2.5 times, this increase in coolant flow rate is due to an increase in hydraulic resistance due to an increase in gas velocity; in the range from 1000 to 1200 rpm, a maximum of the indicator isothermal efficiency and compressor feed rate is observed; maximum indicator isothermal efficiency and the supply ratio is observed at a coolant flow rate in the range from 800 mL/min to 900 mL/min.

Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems

The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L2 as the time step τ tends to zero at the rate O(τ).

Generalized additive-multiplicative risk estimation model based on fuzzy and interval initial data

Among the mathematical models of risk research, an important place is occupied by additive-multiplicative models of risk estimation. The components of such models are: three-stage hierarchical risk systems (built for a specific applied situation); partial risk estimators (determined by experts for a specific project, product, etc.); indicators of the weight of specific types of partial risks (found on the basis of a survey of experts in a particular application area); algorithms for calculating group risk estimators based on partial risk estimators and general risk estimator based on group risk estimators. As examples, three-stage hierarchical risk systems are considered in the production of a new innovative product and in the implementation of projects for the development of rocket and space technology. An algorithm for an additive-multiplicative model for risk estimation of a general form is proposed. Estimates of partial risks are products of weighting indicators by severity indicators, which corresponds to the well-known method of risk estimation in the form of the product of average damage by the probability of an undesirable event. Group risk estimators are built additively from i partial risk estimators, and the final overall risk estimator is calculated multiplicatively from group risk estimators. In previous works of the author, a special case of an additive-multiplicative risk estimation model was considered, in which, in particular, the components of the model were interpreted in terms of probability theory. It is proposed to carry out estimators of partial risks and weight coefficients on the basis of interval mathematics and fuzzy theory. The rules of arithmetic operations on interval and triangular fuzzy numbers are given. The application of the algorithm of the additive-multiplicative risk estimation model based on triangular fuzzy numbers is demonstrated using the example of risk estimation for the implementation of innovative projects. Within the framework of interval mathematics, risk estimators are considered in the implementation of projects for the development of rocket and space technology. The approach developed in this research article corresponds to the main provisions of the theory of stability of mathematical models of real phenomena and processes and to the results of systemic fuzzy interval mathematics.

Online Process State Estimation for Hansenula polymorpha Cultivation with 2D Fluorescence Spectra-Based Chemometric Model Calibrated from a Theoretical Model in Place of Offline Measurements

The use of 2D fluorescence spectra is a powerful, instantaneous, and highly accurate method to estimate the state of bioprocesses. The conventional approach for calibrating a chemometric model from raw spectra needs a large number of offline measurements from numerous runs, which is tedious, time-consuming, and error-prone. In addition, many process variables lack direct signal responses, which forces chemometric models to make predictions based on indirect responses. In order to predict glycerol and biomass concentrations online in batch cultivation of Hansenula polymorpha, this study substituted offline measurements with simulated values. The only data from cultivations needed to generate the chemometric model were the 2D fluorescence spectra, with the presumption that they contain sufficient information to characterize the process state at a measurement point. The remainder of the evaluation was carried out with the aid of a mathematical process model that describes the theoretical interferences between process variables in the system. It is shown that the process model parameters, including microbial growth rate, the yield of biomass from glycerol, and lag time can be determined from only the spectra by employing a model-based calibration (MBC) approach. The prediction errors for glycerol and biomass concentrations were 8.6% and 5.7%, respectively. An improved model-based calibration (IMBC) approach is presented that calibrates a chemometric model for only biomass. Biomass was predicted from a 2D fluorescence spectrum in new cultivations, and glycerol concentration was estimated from the process model utilizing predicted biomass as an input. By using this method, the prediction errors for glycerol and biomass were reduced to 5.2% and 4.7%, respectively. The findings indicate that model-based calibration, which can be carried out with only 2D fluorescence spectra gathered from prior runs, is an effective method for estimating the process state online.

The uniqueness of the solution of a mixed problem for three-dimensional hyperbolic equations with type and order degeneracy property

Abstract The relevance of the stated subject is conditioned upon the presence of a real possibility to simulate vibrations of elastic membranes in space according to the Hamilton principle using degenerate three-dimensional hyperbolic equations, which is of particular practical importance from the standpoint of the prospects for mathematical modelling of the heat propagation process in oscillating elastic membranes. The purpose of this paper is to study the sequence of the procedure for mathematical modelling of heat propagation in oscillating elastic membranes which is leading to degenerate three-dimensional hyperbolic equations. The methodological approach of this study is based on a combination of theoretical study of the possibilities of constructing mathematical models of heat propagation in oscillating elastic membranes with the practical application of methods for constructing three-dimensional hyperbolic equations with type and order degeneracy to find a single solution to a mixed problem. In the course of this study, the results were presented in the form of a mathematical proof of the possibility of obtaining a single solution to a mixed problem for three-dimensional hyperbolic equations with type and order degeneracy. The results obtained in this study and the conclusions formulated on their basis are of significant practical importance for developers of methods of mathematical modelling of heat propagation processes in oscillating artificial membranes, which is of key importance from the standpoint of prospects for improving methods of mathematical modelling of processes occurring in technical devices used in various fields of modern industries.

Measurement of residual stresses after vibroimpulsive hardening of specimen

Subject of research: technique for determining residual stresses of the first kind, in samples that are cut from the spar after vibration shock hardening.
 Purpose of research: to increase the accuracy, as well as to improve the methodology for determining residual stresses.
 Methods and objects of research: the object of the study is residual stresses and the algorithm of their processing. The study used a modified order of approximation of data obtained from sensors.
 Main results of research: a new algorithm for processing experimental data is presented, which allows us to obtain more advanced results of studies of residual stresses in samples. A mathematical model of data processing has been developed to implement the algorithm. This mathematical model for determining residual stresses in the structure has a modified order of approximation of data obtained from sensors. Approximation by a polynomial of the sixth degree occurs before the plot is calculated and constructed, and not after its construction and subsequent smoothing, as was implemented earlier. This technical solution contributes to improving the accuracy and correctness of studies for determining residual stresses.

Influence of temperature and time modes of polymer pipes welding at low temperatures on the structure and properties of materials of compounds. Part I. Calculation of the heat process

Subject of research: thermal process of welding PE100 saddle branches to polyethylene pipes of existing gas pipelines at low temperatures.
 Purpose of research: development of an effective method for welding saddle branches to polyethylene pipes of gas pipelines at ambient temperatures up to minus 50 C based on thermal process control by introducing preheating into the technological process with subsequent cooling to equalize the temperature and cool the joint under a layer of thermal insulation material of calculated thickness.
 Methods and objects of research: in the first part of the work, the object of study is the non-stationary thermal process of welding. The mathematical model of the thermal process takes into account the heat of the phase transformation of polyethylene in the temperature range, the dependence of thermophysical characteristics on temperature and the degree of crystallinity of the material.
 Main results of research: calculations have determined the welding parameters of the saddle branch, which ensure the flow of the thermal process according to the regularity inherent in welding under conditions of permissible ambient temperature. It is expected that the control of the thermal process will make it possible to obtain a structure of the welded joint material that ensures the strength of the seam corresponding to the strength of the seam obtained by welding under conditions of acceptable temperatures.

INFLUENCE OF THE CHEMICAL COMPOSITION OF HIGH-CHROMIUM CAST IRON ON THE CONTENT OF CHROMIUM IN NEAR-CARBIDE ZONES AFTER ANNEALING AT 720 °С

Purpose. Increasing the corrosion resistance of wear-resistant high-chromium materials operating in a liquid abrasive environment makes it possible to increase the service life of products made from these materials, which is an important task in materials science. One of the main reasons for the occurrence of damage during corrosion processes is the uneven distribution of chromium in the metal base, which leads to the formation of microgalvanic pairs on the surfaces of parts in an electrolytic environment. The creation alloys of wear-resistant high-chromium, in the metal base of which microgalvanic vapors are not formed, is a promising direction for increasing the corrosion resistance of such materials. Investigation of chromium distribution processes after heat treatment parts of high-chromium cast iron, namely the formation of metal base zones near carbides.
 Research methods. Analytical review of publications. Metallographic, microstructural and local X-ray microanalysis. Mathematical modeling of processes.
 Results. It was revealed that during the heat treatment in high-chromium cast irons, structures with significant segregation of chromium are formed. It has been established that after annealing of castings at 720 °C, the minimum chromium content in the metal base is observed in the zones near carbides, and the maximum in the central zones of the graids. The presence zones with a chromium content of more than 12 % (positive potential) and less than 12 % (negative potential) leads to the formation of microgalvanic couples, which is the main reason for the acceleration of corrosion damage in the metal base. The dependence of the chromium content in the zones near the carbides after annealing at 720 °C on the chemical composition of cast iron has been determined.
 Scientific novelty. The mathematical model is proposed for determining the chromium content in the zones near carbides after annealing at 720 °C with an exposure of 9 hours from the chemical composition of cast iron in the Fe-C-Cr-Mn-Ni system. This makes it possible to predict the chromium content in the zones near the carbides and prevent the formation of microgalvanic pairs.
 Practical value. The dependence (mathematical model) obtained can be used in the development of compositions of high-chromium wear-resistant cast irons with increased corrosion resistance and machinability.

Methodology of selection and synthesis of mathematical models of the epizootic process

The epizootic process (EP) is a continuous process of the emergence and spread of explicit and latent infections of animals. In some zoonoses, the study of EP by experimental studies can have negative consequences for human health. Therefore, the development of mathematical models of EP is of great practical importance. The purpose of the work is to consider the principles of the methodology for selecting and synthesizing EP models based on the mathematical theory of epidemics (MTE). Of the existing two main types of models are considered - deterministic and stochastic, their advantages and disadvantages are noted in connection with the main parameters used in the modeling of EP. An analytical review of mathematical models of Kermack (W.O.Kermack) and McKendrick; Weiss (George H. Weiss) for the study of EP is presented. Of the existing two main types of models are considered - deterministic and stochastic, their advantages and disadvantages are noted in connection with the main parameters used in the simulation of the EP. The issue of the methodology of selection and synthesis of models of the epizootic process for further forecasting of the most common viral and bacterial diseases of animals using models existing in the mathematical theory of epidemics is considered. The principles of the methodology for the selection and synthesis of MTE models using state graphs and tables of transition intensity models are considered. It is established that the method of selecting a stochastic MTE model for a specific disease or model synthesis uses a table of transition intensity models. It is concluded that the choice between deterministic and stochastic models is determined by the population size, the stage of epizootic/epidemic development, and the requirements for the accuracy of mathematical modeling. The proposed methodology for the selection and synthesis of mathematical models of EP allows you to build a mathematical model for predicting the distress of farms at the regional level. In recent years, there has been a tendency to switch to the use of simulation models and the creation of a bank of mathematical models for predicting epizootic situation indicators based on time series analysis.

METHODS OF MATHEMATICAL SIMULATION AND MACHINE IDENTIFICATION OF ANOMALOUS DIFFUSION PROCESSES

For the class of anomalous diffusion processes, the mathematical models of which are formalized in the form of variational inequalities in partial derivatives, a method of mathematical modeling based on the optimization procedure is proposed. The method is considered in relation to the generalized mathematical model of the studied class of anomalous diffusion processes. Which made it possible to ensure the principle of unification and typification in the application of this method, as well as the correctness of using the generalized mathematical model in applied problems of mathematical modeling of known industrial and practically important natural cases of anomalous diffusion processes. At the same time, the task of implementing mathematical models of anomalous diffusion processes based on the proposed method is reduced to finding the maximum of the Hamiltonian function defined in the state space of the processes under consideration. A method of parametric identification of mathematical models of anomalous diffusion processes in the formulation of the problem of optimal control is also proposed. The method is reduced to the use of the optimization procedure of the gradient projection method. The possibility of solving the problem of parametric identification in cases of both linear and non-linear mathematical models of anomalous diffusion processes is proved. Moreover, the nonlinear formulation of the parametric identification problem does not lead to computational implementation complications, since the solution is based only on finding the gradient projection of the state function of the anomalous diffusion process. The proposed methods are presented in strict compliance with the provisions of functional analysis, which ensures their correctness and adequacy in solving a wide range of applied problems.

Development of a program for mathematical modeling of molecular dynamics processes

The paper presents the results of developing a program for mathematical modeling of molecular dynamics processes in crystal structures using the MatLab software package. The theoretical part describes the differential equations for simulation, their initial and boundary conditions, and the difference approximation. A method was chosen based on the principle of modeling molecular dynamics using one of the pair potentials, namely the Lennard-Jones potential. In the practical part, the chaotic motion (migration) of atoms in 2D and 3D crystal lattices has been simulated. The distribution over the computational cell and the migration of atoms beyond its limits are shown. The dependence between the bound energies in real metals and the computational model has been determined. The potential of interaction has been determined, which turns out to be positive. The developed program showed high accuracy of simulation results, as well as higher speed compared to other similar programs for molecular dynamics simulation.

Mathematical model of collision process of seeder coulter with stones

The article presents the results of scientific-theoretical research on the development of mathematical modeling of the process of collision of the seeder coulter with stones and the introduction of new technologies for sowing seeds of agricultural crops on stony soils to improve the performance of seeding units.

Mathematical modeling of the processes of combined heat and moisture transfer during storage and drying of raw cotton

The article developed a multidimensional mathematical model of joint heat and moisture transfer processes in inhomogeneous porous bodies, considering internal heat and moisture release, heat, and moisture exchange with the environment. Based on the usage of an implicit finite-difference scheme with the second order of precision in time and space variables, an effective numerical solution for resolving issues with heat and moisture transmission has been created. Software was created for studying the processes of heat and moisture transfer during the storage and drying of raw cotton based on the developed numerical algorithm, making it possible to identify and forecast changes in temperature and humidity at arbitrary points of raw cotton with various sizes.

Оптимизация процесса каталитического циклоалкилирования фенола 1-метилциклопентеном

The work presents the results of a study of the cycloalkylation reaction of phenol with 1-methylcyclopentene and the calculation of a regression mathematical model of the process. KN-30 was used as a process catalyst. The effect of input factors (temperature, time, molar ratios of starting components and amount of catalyst) on the yield and selectivity of para-(1-methylcyclopentyl) phenol was studied. As a result, the optimal conditions for obtaining para-(1-methylcyclopentyl) phenol with high yield and selectivity were found. When studying the reaction of cycloalkylation of phenol with 1-methylcyclopentene in the presence of a KH-30 catalyst, it was found that a high yield of the target product of 71.2% (for taken phenol), a selectivity of -92.8% (for the target product) was obtained under the following conditions: temperature - 110 ° C, duration reactions - 5 hours, the molar ratio of phenol to 1-methylcyclopentene 1:1, the amount of catalyst 10% based on the taken phenol. The chemical structure of the synthesized para-(1-methylcyclopentyl) phenol was confirmed by IR, 1Н and 13С NMR methods and its physicochemical parameters were determined. A regression mathematical model of the cycloalkylation process was calculated and the parameters of the optimal mode were found. With the calculated optimal values of the input variables found, a control experiment was set up, which made it possible to find the yield values of the target product Y1=78%, which indicates the acceptability of the developed regression model.

Primena matematičkih metoda i modela u ekonomiji

The development of the mathematical basis of methods for model construction in various fields of science and technology today is very intensive and interdisciplinary. From the perspective of quantitative optimization methods, mathematical models of processes are of the utmost importance, whereby these can be generally represented in the form of linear algebraic equations or equivalent representations. Within the procedure of defining assumptions and adopting a process model, the problem formulation is conducted, whereby this refers to determining the level of problem decomposition and the level of detail in which the problem will be solved. The procedure of mathematical model determination, based on a process model, involves constructing a specific model that shows the relationships between the variables which describe the process, as well as the criterion of the solution effectiveness. The field of operations research that deals with this type of models is mathematical programming. Mathematical programming models represent a class of optimization models due to the fact that their goal is finding optimal solutions to problems. Analogously, the basic feature of today's modern economy is the penetration of mathematical methods into the essence of all economic research. In this context, the main aim of this paper is to point out the growing importance of the application of mathematical techniques, methods and models in economics as a science, as their proper application in economics can reduce many ambiguities in economic theory and practice.

ФОРМУВАННЯ СИСТЕМИ ЕКОНОМІЧНОГО ЗАБЕЗПЕЧЕННЯ СТРАТЕГІЧНОГО РОЗВИТКУ КОКСОХІМІЧНОЇ ПРОМИСЛОВОСТІ

The purpose of the article is the research of existing approaches and the formation of the author's vision of the system of economic support for the strategic development of the domestic coke industry. It was noted that the coke industry is one of the most important branches of the national economy. It has been established that it faces a number of problems and threats, which have led to a significant decrease in production volumes. The need to develop economic support that will allow timely identification of problems, formation and justification of current strategic development priorities in conditions of uncertainty and taking into account existing challenges and threats has been proven. Attention is focused on the variety of approaches to the interpretation of the concept of "economic support" in scientific sources. On the basis of their research, the author's definition of the term "economic support for the strategic development of the coke industry" was formulated. It was noted that the economic support of strategic development is based on the optimal combination of various elements that form a certain system. It is proposed to group them into two blocks: the main one, which is based on the analysis of trends, economic and mathematical modeling of processes and forecasting the development of the industry, and the supporting one. The structural features are determined and the structural scheme of the formation of the system of economic support for the strategic development of the coke-chemical industry of Ukraine is proposed. It has been established that the development of this system is based on certain principles. A list of them is offered and the main content is characterized. It was noted that the formation of a system of economic support for the development of the coke chemical industry will make it possible to reasonably approach the selection or actualization of previously formed strategic vectors of industry development, to focus attention on the most important and choose the most effective methods and tools, to develop and implement systemic strategic measures aimed at the development of this sector.

МЕТОД НЕЛДЕРА-МИДА РЕШЕНИЯ ЗАДАЧИ ОПТИМИЗАЦИИ ГЕОМЕТРИЧЕСКОЙ ФОРМЫ СТВОЛА АВТОМАТИЧЕСКОЙ ПУШКИ ДЛЯ УЛУЧШЕНИЯ КОЛЕБАТЕЛЬНЫХ ХАРАКТЕРИСТИК

The paper presents the formulation of the problem of optimizing the shape of the barrel on the basis of mathematical modeling of the firing process from an automatic 30-mm cannon. The work considers both the standard annular section of the trunk and with stiffeners of two types. The oscillation amplitude was chosen as the objective function of the minimization algorithm. The maximum mass of the trunk and the minimum allowable thickness of the trunk, determined in accordance with the theory of the greatest deformations, were chosen as restrictions. For a standard barrel with an annular cross-section, the external diameters of the barrel were controllable, for barrels with stiffeners past the diameters, the distances to the stiffeners and their size were taken into account. The problem of internal ballistics in the thermodynamic formulation was solved to determine the gas-noise parameters inside the barrel. Muzzle vibrations were determined on the basis of solving the problem of the stress-strain state of the barrel in a one-dimensional formulation, taking into account technological irregularities and the interaction of the projectile with the barrel. The search for the optimal shape of the trunk was carried out by the Nelder-Meade method, restrictions were taken into account using the penalty function method. The results of optimizing the shape of the trunk are presented in graphical and tabular form. The scattering of shells for the barrels of the considered cross-sectional shapes is determined. As a result of optimization, it was possible to increase the accuracy of shooting by more than 2 times.