Modified method of direct in problems of thermal conductivity of annular plates
Keywords:thermal conductivity, dimension reduction, projection method, reduced equations, modified method of lines
In this paper, to solve the initial boundary value problem of thermal conductivity using a numerical-analytical method - a modified method of lines is proposed. The initial equations of thermal conductivity defined in the cylindrical coordinate system are considered in the spatial formulation, which greatly complicates them. As an object on which they are defined, an annular plate is considered, the overall dimensions of which are commensurate. In the problems of calculating of thermal effects in load-bearing elements the first step is to determine the temperature fields, especially if the overall dimensions of the structures are proportional. Such elements include non-thin annular plates. The boundary conditions are considered in a general form too - these are the conditions for convective heat transfer, which using the passage to the limit, turn into boundary conditions of the first and second types. The application of the modified method of lines to reduce the dimensionality of the initial system of equations of nonstationary thermal conductivity used to determine the temperature fields of the load-bearing elements is shown in this paper.
The application of the modified method of lines involves solving these initial boundary value problems in two stages. At the first stage, the dimensionality of the initial equations with respect to variable z is reduced. The Bubnov-Galerkin-Petrov projection method is used to reduce the dimensionality. The so-called functions-"caps" are accepted as basic functions, which are related to the lines plotted on the definition domain of the problem. The projection method is also used to reduce the dimension of the initial and boundary conditions, that allows to formulate a reduced initial-limit problem, which is convenient to solve using the numerical finite-difference method, using explicit or implicit difference schemes. The most successful form of writing the original equations was found, which ensures ease of application of dimensionality reduction of the initial system of equations using a modified method of lines. The calculation took into account the impact of the environment. Reduced equations, boundary and initial conditions are obtained. As a result, the reduced problem has a form convenient to its solution by modern numerical methods.
Kovalenko A. D. Vvedenie v termouprugost' (Introduction to thermoelasticity)/ A. D. Kovalenko. – Kiev: Naukova dumka, 1965. – 204 s.
Karslou G. Teploprovodnost' tverdyh tel (Thermal conductivity of solids)/ G. Karslou, D. Eger. – Moskva: Nauka, 1964. – 488 s.
Marchuk G. I. Vvedenie v proekcionno-setochnye metody (Introduction to projection-grid methods)/ G. I. Marchuk, V. I. Agoshkov. – Moskva: Nauka, 1981. – 416 s.
Modifіkovanij metod pryamih, algoritm jogo zastosuvannya, mozhlivostі ta perspektivi (Modified method of direct, algorithm of its application, possibilities and prospects)/ [V. K. Chibіryakov, A. M. Stankevich, O. P. Koshevij ta іn.]. // Mіstobuduvannya ta teritorіal'ne planuvannya. – 2019. – №70. – S. 595–616.
Chibіryakov V.K. Diskretno-kontinual'na model' dlya rozrahunku tovstih plastin na dinamіchnі vplivi (Discrete-continuous model for calculating thick plates on dynamic effects) / V. K. Chibіryakov, A. M. Stankevich, D. V. Levkіvs'kij. // Mіstobuduvannya ta teritorіal'ne planuvannya. – 2014. – №51. – S. 678–687.
Chibіryakov V. K. Pro odin algoritm rozv`yazannya pochatkovo-granichnih zadach dlya rіvnyannya nestacіonarnoї teploprovіdnostі (About one algorithm for solving initial-boundary value problems for the equation of nonstationary thermal conductivity) / V. K. Chibіryakov, A. M. Stankevich, V. F. Mel'nichuk. // Opіr materіalіv і teorіya sporud. – 2015. – №95. – S. 90–95.
`Application of generalized “method of lines”, for solving problems of thermoelasticity of thick plates. / V.Chybiryakov, A. Stankevich, D. Levkivskiy, V. Melnychuk. // “Motrol”. – 2014. – №8. – С. 11–20.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.