БАҒАЛАУ ӘДІСТЕРІ МЕН АЛГОРИТМДЕРІН ӘЗІРЛЕУ ДЕНЕДЕГІ ТЕМПЕРАТУРАНЫҢ ТАРАЛУЫ ТІКБҰРЫШТЫ ПАРАЛЛЕЛЕПИПЕДТІҢ ПІШІНІ ӘСЕР ЕТЕДІ ЖЫЛУ АҒЫНЫ ЖӘНЕ ЖЫЛУ АЛМАСУДЫҢ БОЛУЫ
DOI:
https://doi.org/10.26577/JMMCS.2023.v117.i1.07Кілттік сөздер:
Вариациялық тәсiл, жылу өткiзгiштiк, жылу ағыны, тiкбұрышты параллелипипедАннотация
Мақалада жылу ағынының әсер етуi және жылу алмасудың болуы кезiнде тiкбұрышты па-
раллелепипед формасындағы денеде температураның таралу Заңын бағалау үшiн әдiстер
мен есептеу алгоритмдерi сипатталған. Жылу ағыны бiр бетке түседi, ал басқа беттер жылу
оқшауланған немесе қоршаған ортаның әсерiнен болады деп саналады. Вариациялық тәсiлдi
қолдану үшiн шекаралық жағдайларды ескере отырып, толық энергияның функционалдығы
есептеледi. Функционалдылықты азайтып, оны нөлге теңестiре отырып, бiз сызықтық тең-
деулер жүйесiн аламыз, оның шешiмi түйiндiк нүктелердегi тiкбұрышты параллелепипедтiң
температурасын бередi. Әрi қарай, температураның түйiндiк мәндерiн жуықтау функция-
сына алмастыра отырып, бiз тiкбұрышты параллелепипед түрiнде денеде температураның
таралу Заңын аламыз. Температураның таралу заңы тiкбұрышты параллелепипедтi бiр, екi және үш элементке бөлу арқылы алынады. Температураның таралу Заңын есептеу процесiн
тездету үшiн алгоритм әзiрлендi, ол есептеу тиiмдiлiгiн реттi арттыратын бағдарлама кодын
жасауға мүмкiндiк бередi. Бұған құрылған кодта негiзгi бағдарламадан айырмашылығы тек
сызықтық теңдеулер жүйесi бар, ол толық энергияның жалпы функционалдығын құрайды,
осы функцияның туындыларын есептейдi және теңдеулер жүйесiн алады.
Библиографиялық сілтемелер
[1] Segerlind J.L., Applied finite element analysis (New York-London-Sydney-Toronto, Jonh Wiley and Song, 1976): 422.
[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.
[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.
Жүктелулер
Как цитировать
Tashev, A., Kazykhan, R., & Aitbayeva, B. (2023). БАҒАЛАУ ӘДІСТЕРІ МЕН АЛГОРИТМДЕРІН ӘЗІРЛЕУ ДЕНЕДЕГІ ТЕМПЕРАТУРАНЫҢ ТАРАЛУЫ ТІКБҰРЫШТЫ ПАРАЛЛЕЛЕПИПЕДТІҢ ПІШІНІ ӘСЕР ЕТЕДІ ЖЫЛУ АҒЫНЫ ЖӘНЕ ЖЫЛУ АЛМАСУДЫҢ БОЛУЫ. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 117(1). https://doi.org/10.26577/JMMCS.2023.v117.i1.07
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