An algorithm of linear combinations: thermal conductivity

V. I. Mashukov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 4, pp. 407-418.

Résumé

This paper presents the computational algorithms, which make it possible to overcome some complexities with the numerical solution of the boundary-value problems of thermal conductivity when the domain of the solution has a complex form or boundary conditions differ from standard ones. Boundary contours are assumed to be broken lines (the flat case) or triangles (a 3D case). Boundary conditions and calculation results are presented as discrete functions whose values or their averaged values are given at geometric centers of boundary elements. Boundary conditions can be defined on the heat flows through boundary elements as well as on temperature, a linear temperature combination and heat flow intensity both at the boundary of the solution domain and inside it. The solution to the boundary value problem is presented in the form of a linear combination of the fundamental solutions of the Laplace equation and their partial derivatives and, also, any solutions of these equations that are regular in the solution domain, the values of functions for which can be calculated at the points of the boundary of the solution domain and at its internal points. If the solution, which participates in the linear combination, is singular, then its average value according to this boundary element is considered.

Bibliographie

[1] Yasnitskii L. N., “Ob odnom sposobe resheniya zadach teorii garmonicheskikh funktsii i lineinoi teorii uprugosti”, Prochnostnye i gidravlicheskie kharakteristiki mashin i konstruktsii, Izd-vo Permskogo politekhnicheskogo instituta, Perm, 1973, 78–83

[2] Yasnitskii L. N., Metod fiktivnykh kanonicheskikh oblastei v mekhanike sploshnykh sred, Nauka, M., 1992

[3] Gladkii S. L., Yasnitskii L. N., “Reshenie trekhmernykh zadach teploprovodnosti metodom fiktivnykh kanonicheskikh oblastei”, Vestnik Permskogo universiteta. Matematika. Mekhanika. Informatika, 2011, no. 1-5, 41–45

[4] Gladkii S. L. Yasnitskii L. N., “Reshenie zadach lineinoi termouprugosti metodom fiktivnykh kanonicheskikh oblastei”, Vestnik Permskogo universiteta. Dinamika i prochnost mashin, 2003, no. 4, 79–90

[5] Shardakov I. N., Metod geometricheskogo pogruzheniya v teorii uprugosti, eds. I. N. Shardakov, N. A. Trufanov, V. P. Matveenko, UrORAN, Ekaterinburg, 1999

[6] Kostyuchenko C. B., “Metod neortogonalnykh ryadov dlya neidealno sopryazhennykh ellipticheskikh zadach s razryvnymi koeffitsientami”, Vychislitelnye tekhnologii, 5:1 (2000), 52–64 | MR | Zbl

[7] Kobylyanskii Yu. V., Popkova T. L., Slepyan G. Ya., “O metode neortogonalnykh ryadov dlya chislennogo resheniya zadach matematicheskoi fiziki”, Zh. vychisl. matem. i mat. fiziki, 28:2 (1988), 237–246 | MR | Zbl

[8] Bolshakov A. Yu., Eltyshev V. A., “O reshenii prostranstvennykh zadach teorii uprugosti metodom Fure”, Staticheskie i dinamicheskie zadachi uprugosti i vyazkouprugosti, Izd-vo UNTs AN SSSR, Sverdlovsk, 1983

[9] Kupradze V. D., Aleksidze M. A., “Metod funktsionalnykh uravnenii dlya priblizhënnogo resheniya nekotorykh granichnykh zadach”, Zh. vychisl. matem. i mat. fiziki, 4:4 (1964), 683–715 | MR | Zbl

[10] Aleksidze M. A., Reshenie granichnykh zadach metodom razlozheniya po neortogonalnym funktsiyam, Nauka, M., 1978

[11] Kupradze V. D., Metody potentsiala v teorii uprugosti, Fizmatgiz, M., 1963

[12] Kupradze V. D., Gegeliya T. G., Basheleishvili M. O., Burchuladze T. V., Trëkhmernye zadachi matematicheskoi teorii uprugosti i termouprugosti, Nauka, M., 1976 | MR

[13] Mashukov V. I., Osinov V. A., “Spektralnaya otsenka tochnosti scheta v zadachakh teorii uprugosti”, Mekhanika tvërdogo tela, 4, 1990, 35–40

[14] Mikhlin S. G., Variatsionnye metody v matematicheskoi fizike, Nauka, M., 1970 | MR

[15] Mashukov V. I., Algoritm lineinykh kombinatsii dlya resheniya granichnykh zadach teorii uprugosti i teploprovodnosti, Dep. v VINITI 31.03.16, No 55-V2016, Sibirskii gosudarstvennyi universitet putei soobscheniya, Novosibirsk, 2016

[16] Mashukov V. I., “An Outer LayerMethod for Solving Boundary Value Problems of Elasticity Theory”, Numerical Analysis and Applications, 10:3 (2017), 237–243 | DOI | DOI | MR | Zbl

[17] Kheigeman L., Yang D., Prikladnye iteratsionnye metody, Mir, M., 1986

[18] Kuznetsov S. B., Algoritm resheniya kvazilineinykh ellipticheskikh uravnenii v oblastyakh s istochnikami na vnutrennikh granitsakh, Preprint 476, AN SSSR. Sib. otd-nie. VTs, Novosibirsk, 1983