Кафедра прикладної математики та обчислювальної техніки (ІПБТ)
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Item Computer Modeling of Harmful Impurities Transfer(Scientific Publishing Center “Sci-conf.com.ua”, 2021) Moroz, Borys Ivanovych; Shvachych, Gennady Grygorovych; Chorna, Valentyna Ivanivna; Voroshylova, Nataliiya VolodymyrivnaENG: The paper considers solutions to the ecology problems, which set is formulated from cause-effect relationships. According to the adopted model, the equation’s coefficients for the harmful impurities transfer are attributed to the causal features of the process. Herein, the setting of cause-and-effect links is the goal of the ecology’s direct problems. Along with direct methods of mathematical modeling of harmful impurities transfer in the atmosphere from pollution sources, the paper considers the formulation and methods of solving inverse problems, which essence is to estimate the input parameters based on actual information about the modeled system, known from the experiment. Based on the research results, a software package was developed to implement the solution of the coefficient inverse problems of ecology using the mathematical modeling method.Item Computer System for Mechanisms Diagnosis(Ukrainian State University of Science and Technologies, Dnipro, 2022) Ivashchenko, Valeriy; Shvachych, Gennady; Sushko, LarysaENG: The computer system proposed in this work is aimed at solving the problem of automating a comprehensive assessment of the technical functioning of mechanisms. The system’s computational equipment have the minimum necessary computing requirements. No additional paid software is required for installation. Unlike existing systems, the proposed one has a moderate cost. For the majority of industrial enterprises, this factor is crucial when choosing the most beneficial computer system. In addition, the developed system is simple and comfortable to use. Thus, the system has an intuitive and intelligible interface for the operator, which allows the operator to quickly familiarize themselves with it and put it to use immediately; the system monitors the correctness entries in the electronic history - it corrects basic fields that are not properly indicated (repair data, repair requests, part price, etc.). The system has the ability to add individual templates for a specific unit. Unlike existing systems, the proposed system is multifunctional.Item Development of a Linear-Scaling Consensus Mechanism of the Distributed Data Ledger Technology(Springer, Singapore, 2022) Shvachych, Gennady G.; Pobochii, Ivan A.; Sashchuk, Hanna; Dzhus, Oleksandr; Khylko, Olena; Busygin, VolodymyrENG: The paper proposes and explores a new blockchain system that operates on a linearly scalable consensus mechanism. This selection method confirms the shard through shares voting and scalable random generation by VDF (Verifiable Delay Function) and VRF (Verifiable Random Function). The system analyzes available consensus mechanisms, sharding, and the age of distributed randomness. It is energy efficient, fully scalable, secure, with fast consensus. Compared to available methods, the improved shard method performs network connection and transaction verification and reveals the state of the blockchain. The threshold has a sufficiently low coefficient for small validators to participate in the network and receive rewards. The proposed sharding process runs securely due to a distributed randomness (DRG) process that is unpredictable, impartial, and verified. The network is constantly overloaded to prevent slow adaptive Byzantine malicious validators. Contrary to other sharding blockchains that require Proof-of-Work to select validators, the proposed consensus is attributed to Proof-of-Stake, therefore, energy-efficient. Herein the consensus is achieved by a BFT algorithm which is linearly scalable and faster than PBFT.Item Efficient Algorithms for Parallelizing Tridiagonal Systems of Equations(НМетАУ, Дніпро, 2021) Shvachych, Gennady Grygorovych; Vozna, Nataliіa; Ivashchenko, Olena Valerievna; Bilyi, Oleksandr; Moroz, DmytroENG: The article is devoted to the development of the maximal parallel forms of mathematical models with a tridiagonal structure. The example of solving the Dirichlet and Neumann problems by the method of straight lines and the sweep method for the heat equation illustrates the direct fundamental features of constructing parallel algorithms. It is noted that the study of the heat and mass transfer processes is run through their numerical modeling based on modern computer technology. It is shown that with the multiprocessor computing systems’ development, there disappear the problems of increasing their peak performance. On the other hand, building such systems, as a rule, requires standard network technologies, mass-produced processors, and free software. The noted circumstances aim at solving the so-called big problems. It should be borne in mind that the classical approach to solving the tridiagonal structure models based on multiprocessor computing systems is far more time-consuming compared to single-processor computing facilities. That is explained by the recurrence relations that make the basis of classical methods. Therefore, the proposed studies are relevant and aim at the distributed algorithms development for solving applied problems. The proposed research aims to construct the maximal parallel forms of mathematical models with a tridiagonal structure. The paper proposes the schemes to implement parallelization algorithms for applied problems and their mapping to parallel computing systems. Parallelization of tridiagonal mathematical models by the method of straight lines and the sweeping method allows designing absolutely stable algorithms with the maximum parallel form and, therefore, the minimum possible time for their implementation on parallel computing devices. It is noteworthy that in the proposed algorithms, the computational errors of the input data are separated from the round-off errors inherent in a PC. The proposed approach can be used in various branches of metallurgical, thermal physics, economics, and ecology problems in the metallurgical industry.Item The Environment Dynamics Identification Based on the Modular Computing Complex(RS Global Sp. z O.O., Poland, 2021) Moroz, Borys; Shvachych, Gennady; Chorna, Valentyna; Voroshylova, NataliiyaENG: The research aims at covering the mathematical modeling issues of multidimensional applied problems of ecology based on the application of a modular computing complex. The problem of modeling air pollution processes is solved by mathematical models that adequately describe fundamental processes. That reveals issues such as a detailed analysis of the atmosphere of the city or industrial area, short-term forecast of air quality in the region, assessment of long term air purification programs, optimal emission management, transboundary transfer, etc. At the same time, the formulation and methods of solving problems of environmental dynamics identification are considered, which essence is to estimate the input parameters based on the factual information about the modeled system known from the experiment. In these studies, the multidimensional equation of harmful impurities transfer was reduced to a sequence of schemes involving unknown values in a single direction, alternately in the longitudinal, transverse and vertical. The implicit schemes lead to systems of algebraic linear equations with a three-diagonal structure. Thus, the methodological basis of the difference splitting schemes provides the economic and sustainable implementation of numerical models by the scalar runs method. That approach focuses on the fact that the greatest effect of a parallel processor is achieved when it is used to perform matrix computations of linear algebra. In order to analyze the feasibility of mathematical models, a package of applications was developed to compute the transfer of harmful impurities. A solution to several applied problems for the identification of the environmental dynamics is given.Item Features of Using the Distance Learning Model(Ukrainian State University of Science and Technologies; Aalto University Helsinki, Finland, 2021) Shvachych, Gennady G.; Khokhlova, Tetiana S.; Stupak, Yurii O.; Moroz, Boris; Kabak, LeonidENG: Purpose. To analyze models of distance learning (DL) that have been implemented in the world and have shown effectiveness in order to determine promising directions for the development of DL: testing, educational research, contextual learning, mobile learning, development of social networks, application of mobile educational games, implementation of mobile voice content with interactive assessment etc. Methodology. Literature review of open access information sources. Findings. It has been determined that special attention should be paid to the technologies of educational institutions, in which it is possible to involve highly qualified teachers in reading lectures; there is a significant saving of teachers' time; there is a possibility of increasing the amount of material presented within the budget allocated for regular classroom activities in the curriculum. Originality. It consists in a certain set of criteria, based on the analysis, which are necessary and sufficient to achieve the final goal. Practical value. The methods of selection of educational tools make it possible to increase the effectiveness of the educational process, make it attractive and exciting for students and motivate them to an active educational process always and everywhere, which allows students to receive a quality education.Item High-Performance Multiprocessor Systems(Український державний університет науки і технологій, ННІ «Інститут промислових та бізнес технологій», 2022) Ivashchenko, Valery Petrovych; Shvachych, Gennady Grygorovych; Ivashchenko, Olena Valerievna; Sushko, Larisa FedorovnaENG: The paper is devoted to the modeling of high-performance multiprocessor architecture for data processing systems used to solve problems with an expandable calculations' area. The proposed system is characterized by high reliability and high energy efficiency. The system contains a separate reconfigurable network for the exchange of data between computing nodes, managed switches. The system also provides network booting nodes and redundancy mechanism of key components.Item Implementation of System Software of Multiprocessor Computing Systems for Solving Applied Tasks(«Пороги», Дніпро, 2021) Shvachych, Gennady Grygorovych; Moroz, D.ENG: The paper proposes a concept for making a multiprocessor computing system designed to solve aviation industry's problems, which would have peak real efficiency and productivity. In addition, this system should be characterized by increased reliability and high energy efficiency. Blocks of the proposed system were implemented with the help of mass production computing facilities. The FAWN technology has further developed. The research considers the problems of use and adjustment of system software of the multiprocessor computing system. For the designed MCS, the OS Linux was selected. It is now considered standard for computing cluster systems. In addition, the section discusses the use of communication tools (interactions) among the nodes of the multiprocessor computing system. It is shown that the most widespread in our time parallel programming technology with partitioned memory is MPI technology. The results of intermediate and final computations are transmitted to the TCA via a managed Infiniband switch. In this case, the control and transfer of the corresponding data from the slave-nodes is carried out by HCA network adapters.Item Intelligent Decision Support System(RS Global Sp. z O.O., Poland, 2021) Shvachych, Gennady; Pobochij, Ivan; Sazonova, Maryna; Bilyi, Oleksandr; Moroz, DmytroENG: Here one of the new efficient approaches to solving problems of intelligent support of making decisions of a cooperate enterprise scale is analyzed, bearing in mind that the enterprise is oriented towards integration of two interconnected information technologies: a technology of constructing information storages and a technology of intelligent data analysis. A new conception of the role of informating-controlling systems in controlling and operating pipe manufacturing process is suggested. It broadens a traditional view over information systems just as if it were only an instrument of a computer data analysis. The role of the modern information systems in the manufacturing as an independent scientific and applied direction serving as a connecting link in the triad “raw product – ngineering process – end product” is substantiated. Some problems connected with plantation of the informating-controlling systems in the pipe manufacturing is formulated. It broadens a traditional view over information systems just as if it were only an instrument of a computer data analysis. The practical application of this product permits the operating personnel of the enterprise to control the engineering process, analyze the processes passing in each mill, adjust the PRP mills’ engineering characteristics in the pipe-manufacturing process. The suggested approach influences quite strongly the production’s quality improving and enables to work stably in the thin-walled pipe grades sphere.Item Intelligent Monitoring and Control System for the Pipe Manufacturing Process(Ukrainian State University of Science and Technologies, Dnipro, 2022) Ivashchenko, Valerii; Shvachych, Gennady Grygorovych; Sobolenko, Oleksandr; Ivashchenko, Olena ValerievnaENG: Here one of the new efficient approaches to solving problems of intelligent support of making decisions of a cooperate enterprise scale is analyzed, bearing in mind that the enterprise is oriented towards integration of two interconnected information technologies: a technology of constructing information storages and a technology of intelligent data analysis. A new conception of the role of informating-controlling systems in controlling and operating pipe manufacturing process is suggested. It broadens a traditional view over information systems just as if it were only an instrument of a computer data analysis. The role of the modern information systems in the manufacturing as an independent scientific and applied direction serving as a connecting link in the triad “raw product – ngineering process – end product” is substantiated. Some problems connected with plantation of the informating-controlling systems in the pipe manufacturing is formulated. It broadens a traditional view over information systems just as if it were only an instrument of a computer data analysis.Item Main Mechanisms of Blockchain Technology Implementation in Digital Technologies Application(Dnipro University of Technology, Dnipro, Ukraine, 2022) Shvachych, Gennady Grygorovych; Moroz, B. І.; Pobochii, Ivan A.; Timchenko, О. P.; Kozenkova, V. D.; Busygin, V. V.ENG: Purpose. To analyze the basic principles of blockchain technology implementation, highlighting the algorithms for reaching consensus in the blockchain network to ensure its reliability; to identify key problems in the implementation of such technology and suggest ways to overcome them; to perform a systematic analysis of the blockchain technology contradictions and suggest ways to eliminate them. Methodology. The research used the basics of economic analysis of economic entities to compare their management’s centralized and decentralized models. This approach showed that another wave of transformation of business and social models has unfolded in recent years, caused by the next-generation digital technologies involving the economy’s transition to the digital area. Revealing the blockchain technology mechanisms is based on modern databases and peer-to-peer computer networks, covering in detail the main means of contradictions, application, and implementation of blockchain technology. Findings. The main results of these studies are obtained in the digital economy. The paper shows that digital technologies open up a wide range of opportunities for different sectors of the economy. The research highlights the features and principles of distributed registry technology (Blockchain) applications. It is shown that as a decentralized data registry, blockchain technology is themost discussed and relevant topic in the digital economy. Originality. The paper further developed the main component of the digital economy, which is progressing most intensively, – the distributed ledger technology (Blockchain). The paper analyzed its strengths, such as cost reduction, increased security, and transparency of transactions that attracted the attention of various sectors of the economy. The authors’ approach for eliminating the revealed mechanisms of contradictions, application, and implementation of blockchain technology is presented. The definition of the digital economy, digital technologies in the economy, and “end-to-end” digital technologies have been improved to clarify the understanding of the economic management decentralization problems. That showed that the digital economy has several subtleties associated with insufficient research and comprehension of technical implementation and flexibility. Practical value. The research results will be useful for expanding ideas about the blockchain technology implementation in different sectors of the economy, accompanied by lower costs, increased security and transparency of economic entities, and improving their economic efficiency and development in digital technologies’ application. The blockchain technologies implantation at the enterprises of the mining and metallurgical industry allows making the production and sales of products more efficient and transparent, and at the same time significantly reduces the human factor.Item Main Problems and Principles of Implementation of Distributed Data Registry Technology(Scientific Publishing Center “Sci-conf.com.ua”, 2022) Shvachych, Gennady Grygorovych; Kholod, Olena Grygorivna; Chernetskyi, Sergiy OleksandrovichENG: This paper shows that information and communication technologies, particularly the digital economy, reveal a huge range of opportunities. It was noted that its strengths, the reducing costs, increasing the level of security, and transparency of transactions drew the attention of various economic sectors. In this regard, for a clear and definite understanding of the problems under consideration, the author's definition of the digital economy, digital technologies in the economy, and "end-to-end" digital technologies in the economy were introduced. The paper highlights the most intensively developing main components of the digital economy: distributed ledger technologies (blockchain) and real option technologies. Research has shown that blockchain technology is one of the main directions for the digital economy development. As a decentralized data registry, the paper shows that blockchain technology is the most discussed and relevant topic in the digital economy. Its strengths are analyzed, such as cost reduction, high security, and transaction transparency applicable to various economic sectors. The main mechanisms of contradictions, arrangement, and implementation of blockchain technology are detailed with the author's approach to eliminating the identified contradictions.Item Method of Lines in Distributed Problems of Experimental Data Processing(RS Global Sp. z O.O., Poland, 2021) Shvachych, Gennady Grygorovych; Vozna, Nataliіa; Ivashchenko, Olena Valerievna; Bilyi, Oleksandr; Moroz, DmytroENG: In many cases, the mathematical support of non-stationary thermal experiments is based on methods for solving the inverse heat conduction problem (IHCP), which include boundary thermal conditions determination, identification of heat and mass transfer processes, restoration of external and internal temperature fields, etc. However, at present, the main field of the IHCP application remains the processing and interpretation of the results of the thermal experiments. It was here where the most considerable theoretical and applied successes were achieved in methods' effectiveness and the breadth of their practical use. This paper highlights the issues of mathematical modeling of multidimensional non-stationary problems of metallurgical thermophysics. The primary research purpose aims at solving problems associated with identifying parallel structures of algorithms and programs and their reflection in the computers’ architecture in solving a wide range of applied problems. Supercomputers are currently inaccessible due to the enormous cost and service price. In this regard, a real alternative is cluster-type computing systems by which the simulation results are covered in this paper. Being a relatively new technology, cluster-type parallel computing systems are useful in solving a large class of non-stationary multidimensional problems, while allowing to increase the productivity and quality of computations. The software developed in this paper can be used to plan and process the results of a thermophysical experiment. The algorithms developed in the application program package are simply reconstructed to solve other coefficient and boundary problems of thermal conductivity. The developed algorithms for solving thermophysical problems are highly accurate and efficient: the test solution for IHCP with accurate input data coincides with the thermophysical features of the sample material. The developed software for processing the results of a thermophysical experiment is self-regulating. Moreover, it is quite merely tuned to the solution of others and, in particular, of boundary IHCP.Item Object-Oriented Model of Constructing of the Distributed Databases(ТОВ «ЮРИДИЧНА КОМПАНІЯ «ЮРСЕРВІС», 2022) Shvachych, Gennady Grygorovych; Moroz, B.; Hulina, I.ENG: The proposed object-oriented system model provides a solution to a complex of planning and reporting problems in fuzzy sets based on symmetric pairs of cryptographic keys. In the proposed system, the object identifiers (cryptographic keys) describing structure of the high educational establishments and the learning process are automatically generated on the basis of complete (standardized) names of structural divisions, specialties, disciplines, etc. Unlike classic relational databases, the object-oriented model allows solving effectively problems of such automatic transactions, such as inheritance of the past period plan parameters in the new period plan, the inheritance of the previously described structures. In addition, it allows removal and adding to the work plan of new periods, as well as new structures designed to provide interactive automatic postings of students during the entire period of their studying, etc.Item Research of Computing Efficiency in Modular Multiprocessor Systems(НМетАУ, Дніпро, 2021) Ivashchenko, Valery Petrovych; Shvachych, Gennady Grygorovych; Ivashchenko, Olena ValerievnaENG: The article is devoted to the research of efficiency of a multiprocessor computing system in solving problems aimed at expanding the computing area. The purpose of the research is to further develop the approach associated with definition of methodology for effectiveness evaluation of the multiprocessor modular computing system and the impact on this indicator of the calculations deceleration. At the same time, the main attention is paid to the impact peculiarities on this indicator of network interface of multiprocessor system. We should note that the research of these problems is important and relevant. This is not only due to the fundamental limitation of the maximum possible performance of ordinary serial computers, but also to the almost constant existence of computational problems which solution requires additional capabilities of existing computer facilities. The working methods of analyzing the multiprocessor systems efficiency do not allow to determine the optimal number of its nodes for solving a certain class of problems taking into account the calculations deceleration. At the same time, to solution of such problems is short of proper research development on the analysis of the network interface impact on the efficiency of modular multiprocessor computer systems. In addition, to estimate the efficiency of a computing multiprocessor system, the basic analytical relationships are usually not provided through the parameters of the system being studied. The basic regularities concerning the time of solving the problem are revealed, depending on the change in the multiprocessor system calculations area. The research is aimed at determining the deceleration factor associated with the increase of the computing area of a multiprocessor system when compared with the computer version with an unlimited computing area. The analytical ratios are derived for determining the calculations deceleration coefficient. A stage of simulation for calculations of the deceleration factor was carried out to determine the regularities of its change, depending on the application of a particular computing platform. The revealed tendencies of such a change point to the need to reconcile the components of the network interface and computing capabilities of the chosen computing platform. The derived analytical relations were aimed at determining the optimal number of nodes of a multiprocessor system which allow the minimum delay of calculations.Item Rolling Process Decision Making Support System(Scientific Publishing Center “Sci-conf.com.ua”, 2021) Ivashchenko, Valery Petrovych; Shvachych, Gennady Grygorovych; Ivashchenko, Olena Valerievna; Zaporozhchenko, Olena EugeneENG: Here one of the new efficient approaches to solving problems of intelligent support of making decisions of a cooperate enterprise scale is analyzed, bearing in mind that the enterprise is oriented towards integration of two interconnected information technologies: a technology of constructing information storages and a technology of intelligent data analysis. A new conception of the role of informating-controlling systems in controlling and operating pipe manufacturing process is suggested. It broadens a traditional view over information systems just as if it were only an instrument of a computer data analysis.Item Some Complex Intensification Features of Spheroidizing Annealing of Low Carbon Steel(Hrvatsko Metalursko Društvo (HMD), Zagreb, Croatia, 2022) Shvachych, Gennady Grygorovych; Mamuzić, I.; Tsvykh, V.; Khуlko, M.; Sashchuk, H.; Timchenko, O.; Ivashchenko, Olena ValerievnaENG: The paper considers complex intensification features of spheroidizing annealing of low-carbon steels and possible technological realizations of intensive annealing modes in current lines. The research aims to reveal the intensification nature of the steel’s spheroidizing due to the non-isothermal holding and an internal coolant for the metal heating. That allows a significant reduction of the spheroidizing annealing process while improving the steel product’s technological properties – providing a high dispersion and homogeneity of the structure across the entire plane of its section. The multiprocessor computing system with its mathematical and IT software for modeling the heat treatment modes of metal billets effectively controls the processes.Item Дослідження успішності студентів з проблемами слуху(НМетАУ, Дніпро, 2021) Дорош, Наталія Леонідівна; Астіон, Юлія Володимирівна; Петречук, Ліна МиколаївнаUKR: Проведено дослідження загальних показників та успішності студентів, які відносяться до Регіонального центру освіти інвалідів Національної Металургійної академії України (РЦОІ). Приймаючи до уваги, що Металургійна академія є єдиним закладом IV рівня акредитації в Україні, у якому здобувають освіту студенти з проблемами слуху, надана робота може бути віднесена до актуальних питань сьогодення.. Проведено збір та обробку інформації з навчання студентів з проблемами слуху з 2007 по 2018 роки. Спроектовано і розроблено базу даних РЦОІ. Проведено дослідження якості успішності студентів з проблемами слуху, які навчалися за спеціальністю «Комп’ютерні науки». Виконано аналіз пріоритетності дисциплін в їх освоєнні. Запропоновано об’єднання дисциплін за 6 категоріями. Доведено, що всі категорії дисциплін засвоюються, але дисципліни гуманітарної та соціальної спрямованості засвоюються легше. Підтверджено ефективність та доцільність отримання вищої освіти студентами з проблемами слуху.Item Математичне моделювання в задачах геометрично нелінійної теорії пружності(Херсонський національний технічний університет, 2021) Кагадій, Тетяна Станіславівна; Шпорта, Анна Григорівна; Білова, Оксана Вікторівна; Щербина, Ірина ВолодимирівнаUKR: Розв’язки багатьох важливих для практики задач, що виникають в сучасній техніці, не завжди можуть бути отримані традиційними методами теорії аналітичних функцій або за допомогою інтегральних перетворень. Це відноситься, наприклад, до контактних задач, в яких враховуються скінченні розміри області хоча б в одному напрямку, або досліджуються середовища з криволінійною анізотропією тощо. Засоби математичної теорії пружності виявляються не надто ефективними для дослідження таких задач. У цьому випадку доцільно використовувати досягнення теорії потенціалу. Застосування ж асимптотичних методів при цьому, навіть в складних випадках, дозволяє отримувати обґрунтовані наближені рівняння, уточнювати якісні закономірності і отримувати аналітичні розв’язки задач. У даній роботі представлене узагальнення методу збурень, яке дозволяє звести дослідження складних задач геометрично нелінійної теорії пружності (в плоскій та просторовій постановці) до послідовного розв’язання більш простих крайових задач теорії потенціалу. Геометрично нелінійна теорія пружності містить в собі деякі особливості, завдяки яким вона відрізняється від класичної (лінійної) теорії. Головна відмінність полягає в урахуванні різниці між геометрією недеформованого та деформованого станів досліджуваного тіла, коли мають місце переміщення, які викликають значні зміни геометрії тіла. При цьому рівняння рівноваги необхідно складати з урахуванням зміни форми і розмірів конструкцій. Врахування кінцевих деформацій, які при створенні математичних моделей веде до значних труднощів при розв’язуванні задач, але в той же час наближає модель до реальної проблеми. Метод збурень, що використовується для розв’язання нелінійних рівнянь у частинних похідних, має теоретичне і практичне значення. Він універсальний і може використовуватися для аналізу різних завдань математичної фізики. Розроблений підхід може бути застосований для вирішення завдань, в яких істотну роль грають залишкові деформації. Наприклад, згин тонких пластин і оболонок. У розглянутій модельній задачі вдалося виділити вплив геометричної нелінійності на напружено-деформований стан досліджуваного тіла. Саме тому результати представленої роботи мають як теоретичне, так і прикладне значення, а дослідження є актуальним.Item Математичне моделювання при дослідженні напружено-деформованого стану smart-матеріалів(ФОП Іванченко І. С., Харків, 2022) Кагадій, Тетяна Станіславівна; Білова, Оксана Вікторівна; Шпорта, Анна Григорівна; Онопрієнко, Олег ДмитровичUKR: Активні матеріали, перш за все п'єзоелектричні і пьєзоелектромагнітні, часто використовуються в якості функціональних частин різних електронних пристроїв, включаючи датчики та перетворювачі, оскільки ці матеріали здатні змінювати свою форму під дією електричного або магнітного поля. У багатьох випадках розміри згаданих пристроїв надзвичайно малі, але тим не менше вони можуть піддаватися впливу дуже великих механічних, електричних і магнітних полів. Крім того, ці пристрої зазвичай складаються з елементів, що можуть бути виготовлені з різних матеріалів (п'єзоелектричні або пьєзоелектромагнітні елементи, електроди тощо). Дослідження поведінки конструкцій, виготовлених з таких матеріалів, виявляє суттєві математичні труднощі під час відповідних розрахунків. Необхідність у вирішенні цих питань, яка проявилася на практиці, зумовила важливість розробки методів розрахунку контактних взаємодій, а також дослідження контактних задач з урахуванням п'єзоелектричної і п'єзоелектромагнітної складової. Проведено узагальнення методу малого параметру на двовимірні задачі електропружності. Перевагою запропонованого узагальненого методу збурень є те, що він дозволяє звести розв’язання складних задач теорії електропружності до послідовного розв’язання більш простих крайових задач (інтегрування рівнянь Лапласа, у складніших випадках – рівняння Пуассона). Доведено, що в усіх випадках вихідної постановки можуть бути сформульовані крайові умови для основних функцій. Механічні та електричні складові можуть бути відокремлені, але мають взаємний вплив через крайові умови. Розв’язок знаходиться як суперпозиція розв’язків для кожного напруженого стану.