РАЗРАБОТКА МЕТОДОВ И АЛГОРИТМОВ ОЦЕНКИ РАСПРЕДЕЛЕНИЕ ТЕМПЕРАТУРЫ В ТЕЛЕ ФОРМА ПРЯМОУГОЛЬНОГО ПАРАЛЛЕЛЕПИПЕДА ПОД ВОЗДЕЙСТВИЕМ ТЕПЛОВОГО ПОТОКА И НАЛИЧИЯ ТЕПЛООБМЕНА
DOI:
https://doi.org/10.26577/JMMCS.2023.v117.i1.07Ключевые слова:
вариационный подход, теплопроводность, тепловой поток, прямоугольный параллелепипед, теплообмен, температураАннотация
В статье описаны методы и вычислительные алгоритмы для оценки закона распределения
температуры в теле формы прямоугольного параллелепипеда при воздействии теплового
потока и наличия теплообмена. Считается, что на одно из грани поддается тепловой поток,
а другие грани теплоизолированные или находятся под воздействием окружающей среды.
Для использования вариационного подхода вычисляется функционал полной энергии,
учитывающий граничные условия. Минимизируя функционал и приравнивая его к нулю
получаем систему линейных уравнений, решение которой дает температуру прямоугольного
параллелепипеда в узловых точках. Далее, подставляя эти узловые значения температур
в аппроксимирующую функцию, получим закон распределения температуры в теле форме
прямоугольного параллелепипеда. Закона распределения температуры получена при
разбиения прямоугольного параллелепипеда на один, два и три элемента. Для ускорения
процесса вычисления закона распределения температуры разработан алгоритм, позволяю-
щий создать код программы, который увеличивают эффективность вычисления на порядок.
Это достигается тем, что создаваемый код содержит только систему линейных уравнений в
отличии от основной программы которая формирует общую функционал полной энергий,
вычисляет производные от этого функционала и получает систему уравнений.
Библиографические ссылки
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[4] Arshidinova М., Begaliyeva К., Kudaykulov А., Tashev А., “Numerical Modeling оf Nonlinear Thermomechanical Processes in a rod of variable cross section in the presence of heat flow”, International Conference on Information Science and Control Engineering (2018): 351–354.
[5] Beck J.V., “Verification solution for partial heating of rectangular solid”, International Journal of Heat and Mass Transfer (2004): 4243–4255.
[6] Beck J.V., “Cole K. D. Improving convergence of summations in heat conduction”, International Journal of Heat and Mass Transfer (2007): 257–268.
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[9] Kraynov A.Yu., Osnovyi teploperedachi. Teploperedacha cherez sloy veschestva: ucheb. posobie [Fundamentals of heat transfer. Heat transfer through a layer of matter: textbook] (Tomsk, 2016): 48.
[10] Gao F., Yang X.J., “Local fractional Euler’s method for the steady heat-conduction problem”, Thermal Science (2016): 735–738.
[11] Petrova L.S., “Matematicheskoe modelirovanie protsessov nagreva kusochno-odnorodnyih tel s uchetom relaksatsii teplovogo potok [Mathematical modeling of heating processes of piecewise homogeneous bodies, taking into account the relaxation of the heat flow]”, Internet-zhurnal «Naukovedenie» 9(1) (2017).
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[13] Ovchinnikov S.V., Teploprovodnost v pryamougolnyih tverdyih telah s vnutrennim vyideleniem tepla: ucheb. dlya vuzov [Thermal conductivity in rectangular solids with internal heat release: textbook] (M.: SGU imeni N.G. Chernyishevskogo,2015): 104.
[14] Tsega E.G., “Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates”, Journal of Applied Mathematics (2022): 8.
[15] Galanin M. P., Proshunin N.N., Rodin A.S., Sorokin D.L., “Reshenie trehmernogo nestatsionarnogo uravneniya
teploprovodnosti metodom konechnyih elementov s uchetom fazovyih perehodov [Solving the three-dimensional nonstationary heat equation by the finite element method, taking into account phase transitions]”, Preprintyi IPM im. M.V.Keldyisha 27(66) (2016). DOI:10.20948/prepr-2016-66.
[16] Goloviznin V.M., Koterov V.N., Krivtsov V.M., “Raschet uravneniya teploprovodnosti na nestrukturirovannyih
krivolineiynyih setkah [Calculation of the heat equation on unstructured curvilinear meshes]”, Zhurnal vyichislitelnoy matematiki i matematicheskoy fiziki (2011): 2075–2083.
[17] Brata S., Maduta C., Pescari S., “Steady-State Three - Dimensional Numerical Simulation of Heat Transfer for Thermal Bridges Assessment”, Politehnica University of Timisoara, Faculty of Civil Engineering, Civil Engineering and Services Department. Journal of applied engineering sciences (2016): 17–22.
[18] Zhalnin R.V., Ladonkina M.E., “Reshenie trehmernyih uravneniy teploprovodnosti s pomoschyu razryivnogo metoda Galerkina na nestrukturirovannyih setkah [Solving 3D Heat Equations Using the Discontinuous Galerkin Method on Unstructured Meshes]”, Vestn. Samarskiy gosudarstvennyiy universitet. Ser. Fiziko-matematicheskie nauki 19(3) (2015): 523–533.
[19] Povstenko Y., Kyrylych T., “Fractional heat conduction in solids connected by thin intermediate layer”, Continuum Mechanics and Thermodynamics (2019): 1719–1731.
[20] Dozhdikov V.I., Poryadin S.V., Dozhdikov K.V., “Optimizatsiya upravleniya protsessom teploprovodnosti v tverdom tele [Optimization of the control of the heat conduction process in a solid]”, Vestn. TGU 4(14) (2009): 704–705.
[21] Ivanov V.V., Karaseva L.V., “Odin iz vozmozhnyih variantov priblizhennogo resheniya zadach nelineynoy teploprovodnosti [One of the possible options for the approximate solution of problems of nonlinear heat conduction]”, Vestn. Inzhenernyiy Dona 2 (2019).
[22] Pavlov V.P., Kudoyarova V.M., “Analiz temperaturnogo polya v tverdom tele metodom splaynov [Analysis of the
temperature field in a solid by the spline method]”, Vestn. Ugatu 22(2) (2018): 10–17.
[23] Kutateladze S.S., Osnovyi teorii teploobmena [Fundamentals of the theory of heat transfer] (M.: Atomizdat, 1979): 416.
[24] Lyikov A.V., Teoriya teploprovodnosti: ucheb. posobie [Theory of thermal conductivity: textbook] (M.: Vyisshaya shkola, 1967): 600.
[25] Haji-Sheikh A., Beck J.V., “Temperature solution in multi-dimensional multi-layer bodies”, International Journal of Heat and Mass Transfer (2002): 1865–1877.
[26] Carslaw H.S., Jaeger J.C., Conduction of heat in solids. Second edition (Oxford clarendon press, 1959).
[27] Rogi´e B., “Practical analytical modeling of 3D multi-layer Printed Wired Board with buried volumetric heating sources”, International Journal of Thermal Sciences (2018): 404–415.
[2] Zenkeevich O., Morgan K., Konechnyie elementyi i approksimatsiya [Finite elements and approximation] (M.: Mir, 1986): 318.
[3] Kudaykulov A., Tashev A.A., Askarova A., “Computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section”, From the journal Open Engineering (2018): 170–175.
[4] Arshidinova М., Begaliyeva К., Kudaykulov А., Tashev А., “Numerical Modeling оf Nonlinear Thermomechanical Processes in a rod of variable cross section in the presence of heat flow”, International Conference on Information Science and Control Engineering (2018): 351–354.
[5] Beck J.V., “Verification solution for partial heating of rectangular solid”, International Journal of Heat and Mass Transfer (2004): 4243–4255.
[6] Beck J.V., “Cole K. D. Improving convergence of summations in heat conduction”, International Journal of Heat and Mass Transfer (2007): 257–268.
[7] Wang X.Y., “Local fractional functional decomposition method for solving local fractional Poisson equation in steady heat-conduction problem”, Thermal Science (2016): 785–788.
[8] Isachenko V.P., Osipova V.A., Sukomel A.S., Teploperedacha: ucheb. dlya vuzov [Heat transfer: textbook for universities] (M.: Energiya 1975): 488.
[9] Kraynov A.Yu., Osnovyi teploperedachi. Teploperedacha cherez sloy veschestva: ucheb. posobie [Fundamentals of heat transfer. Heat transfer through a layer of matter: textbook] (Tomsk, 2016): 48.
[10] Gao F., Yang X.J., “Local fractional Euler’s method for the steady heat-conduction problem”, Thermal Science (2016): 735–738.
[11] Petrova L.S., “Matematicheskoe modelirovanie protsessov nagreva kusochno-odnorodnyih tel s uchetom relaksatsii teplovogo potok [Mathematical modeling of heating processes of piecewise homogeneous bodies, taking into account the relaxation of the heat flow]”, Internet-zhurnal «Naukovedenie» 9(1) (2017).
[12] Yilmazer A., Kocar C., “Heat conduction in convectively cooled eccentric spherical annuli: A boundary integral moment method”, Thermal Science (2017): 2255–2266.
[13] Ovchinnikov S.V., Teploprovodnost v pryamougolnyih tverdyih telah s vnutrennim vyideleniem tepla: ucheb. dlya vuzov [Thermal conductivity in rectangular solids with internal heat release: textbook] (M.: SGU imeni N.G. Chernyishevskogo,2015): 104.
[14] Tsega E.G., “Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates”, Journal of Applied Mathematics (2022): 8.
[15] Galanin M. P., Proshunin N.N., Rodin A.S., Sorokin D.L., “Reshenie trehmernogo nestatsionarnogo uravneniya
teploprovodnosti metodom konechnyih elementov s uchetom fazovyih perehodov [Solving the three-dimensional nonstationary heat equation by the finite element method, taking into account phase transitions]”, Preprintyi IPM im. M.V.Keldyisha 27(66) (2016). DOI:10.20948/prepr-2016-66.
[16] Goloviznin V.M., Koterov V.N., Krivtsov V.M., “Raschet uravneniya teploprovodnosti na nestrukturirovannyih
krivolineiynyih setkah [Calculation of the heat equation on unstructured curvilinear meshes]”, Zhurnal vyichislitelnoy matematiki i matematicheskoy fiziki (2011): 2075–2083.
[17] Brata S., Maduta C., Pescari S., “Steady-State Three - Dimensional Numerical Simulation of Heat Transfer for Thermal Bridges Assessment”, Politehnica University of Timisoara, Faculty of Civil Engineering, Civil Engineering and Services Department. Journal of applied engineering sciences (2016): 17–22.
[18] Zhalnin R.V., Ladonkina M.E., “Reshenie trehmernyih uravneniy teploprovodnosti s pomoschyu razryivnogo metoda Galerkina na nestrukturirovannyih setkah [Solving 3D Heat Equations Using the Discontinuous Galerkin Method on Unstructured Meshes]”, Vestn. Samarskiy gosudarstvennyiy universitet. Ser. Fiziko-matematicheskie nauki 19(3) (2015): 523–533.
[19] Povstenko Y., Kyrylych T., “Fractional heat conduction in solids connected by thin intermediate layer”, Continuum Mechanics and Thermodynamics (2019): 1719–1731.
[20] Dozhdikov V.I., Poryadin S.V., Dozhdikov K.V., “Optimizatsiya upravleniya protsessom teploprovodnosti v tverdom tele [Optimization of the control of the heat conduction process in a solid]”, Vestn. TGU 4(14) (2009): 704–705.
[21] Ivanov V.V., Karaseva L.V., “Odin iz vozmozhnyih variantov priblizhennogo resheniya zadach nelineynoy teploprovodnosti [One of the possible options for the approximate solution of problems of nonlinear heat conduction]”, Vestn. Inzhenernyiy Dona 2 (2019).
[22] Pavlov V.P., Kudoyarova V.M., “Analiz temperaturnogo polya v tverdom tele metodom splaynov [Analysis of the
temperature field in a solid by the spline method]”, Vestn. Ugatu 22(2) (2018): 10–17.
[23] Kutateladze S.S., Osnovyi teorii teploobmena [Fundamentals of the theory of heat transfer] (M.: Atomizdat, 1979): 416.
[24] Lyikov A.V., Teoriya teploprovodnosti: ucheb. posobie [Theory of thermal conductivity: textbook] (M.: Vyisshaya shkola, 1967): 600.
[25] Haji-Sheikh A., Beck J.V., “Temperature solution in multi-dimensional multi-layer bodies”, International Journal of Heat and Mass Transfer (2002): 1865–1877.
[26] Carslaw H.S., Jaeger J.C., Conduction of heat in solids. Second edition (Oxford clarendon press, 1959).
[27] Rogi´e B., “Practical analytical modeling of 3D multi-layer Printed Wired Board with buried volumetric heating sources”, International Journal of Thermal Sciences (2018): 404–415.
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Как цитировать
Tashev, A., Kazykhan, R., & Aitbayeva, B. (2023). РАЗРАБОТКА МЕТОДОВ И АЛГОРИТМОВ ОЦЕНКИ РАСПРЕДЕЛЕНИЕ ТЕМПЕРАТУРЫ В ТЕЛЕ ФОРМА ПРЯМОУГОЛЬНОГО ПАРАЛЛЕЛЕПИПЕДА ПОД ВОЗДЕЙСТВИЕМ ТЕПЛОВОГО ПОТОКА И НАЛИЧИЯ ТЕПЛООБМЕНА. Вестник КазНУ. Серия математика, механика, информатика, 117(1). https://doi.org/10.26577/JMMCS.2023.v117.i1.07
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Прикладная математика
