On the joint application of the collocation boundary element method and the Fourier method for solving problems of heat conduction in finite cylinders with smooth directrices

Ivanov D.Yu.

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 72 (2021), pp. 15-38

Résumé

The solution of heat conduction problems in a straight cylinder with zero boundary conditions on the bases and zero initial condition is investigated using the combined use of the collocation method of boundary elements and the Fourier method. Due to the moderate mesh refinement, which compensates for the drop in accuracy for large eigenvalues of the differential operator $\partial^{2}_{уу}$ with the corresponding zero boundary conditions, approximate solutions are obtained that stably converge to exact ones with a cubic velocity uniformly with respect to the generator length and uniformly with respect to the sets of boundary functions bounded in the norm of functions with low smoothness in the variable $y$. The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder.

Détail

Keywords: unsteady heat conduction, boundary integral equations, boundary element, collocation, stability, non-circular cylinder, Fourier method, dissipation.
Mots-clés : uniform convergence

Bibliographie

[1] Brebbiya K., Telles Zh., Vroubel L., Metody granichnykh elementov, Mir, M., 1987, 524 pp.

[2] Onishi K., “Convergence in the boundary element method for heat equation”, Teaching for Robust Understanding of Mathematics, 17 (1981), 213–225

[3] Costabel M., Onishi K., Wendland W.L., “A boundary element collocation method for the Neumann problem of the heat equation”, Inverse and Ill-Posed Problems, eds. H.W. Engl, C.W.Groetsch, Academic Press, Boston, 1987, 369–384

[4] Iso Y., Takahashi S., Onishi K., “Numerical convergence of the boundary solutions in transient heat conduction problems”, Topics in Boundary Element Research, v. 3, ed. C.A. Brebbia, Springer, Berlin, 1987, 1–24

[5] Hongtao Y., “On the convergence of boundary element methods for initial-Neumann problems for the heat equation”, Mathematics of Computation, 68:226 (1999), 547–557

[6] Majchrzak E., Ladyga E., Mendakiewicz J., Belkhayat A.P., “Different variants of the boundary element method for parabolic equations”, Journal of Applied Mathematics and Computational Mechanics, 3:1 (2004), 127–132

[7] Werner-Juszczuk A.J., Sorko S.A., “Application of boundary element method to solution of transient heat conduction”, Acta Mechanica et Automatica, 6:4 (2012), 67–74

[8] Kukla S., Siedlecka U., “Heat conduction problem in a two-layered hollow cylinder by using the green's function method”, Journal of Applied Mathematics and Computational Mechanics, 12:2 (2013), 45–50 | DOI

[9] Kukla S., Siedlecka U., “Green's function for heat conduction problems in a multi-layered hollow cylinder”, Journal of Applied Mathematics and Computational Mechanics, 13:3 (2014), 115–122 | DOI

[10] Pettres R., Lacerda L.A., Carrer J.A.M., “A boundary element formulation for the heat equation with dissipative and heat generation terms”, Engineering Analysis with Boundary Elements, 51, February (2015), 191–198 | DOI

[11] Yang D.-S., Ling J., “A new boundary-type mesh-free method for solving the multi-domain heat conduction problem”, Numerical Heat Transfer, Part B: Fundamentals, 69:2 (2016), 167–178 | DOI

[12] Godinho L., Tadeu A., Simoes N., “Study of transient heat conduction in 2.5D domains using the boundary element method”, Engineering Analysis with Boundary Elements, 28 (2004), 593–606 | DOI

[13] Lu X., Tervola P., Viljanen M., “Transient analytical solution to heat conduction in composite-circular cylinder”, International Journal of Heat and Mass Transfer, 49 (2006), 341–348 | DOI

[14] Asgari M., Akhlaghi M., “Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length”, Heat Mass Transfer, 45 (2009), 1383–1392 | DOI

[15] Gonzalez-Duran J.E.E., Rodriguez-Resendiz J., Ramirez J.M.O., Zamora-Antunano M.A., Lira-Cortes L., “Finite-Element Simulation for Thermal Modeling of a Cell in an Adiabatic Calorimeter”, Energies, 13:9 (2020), 2300, 12 pp. | DOI

[16] Nolasco C., Jacome N.J., Hurtado-Lugo N.A., “Solution by numerical methods of the heat equation in engineering applications. A case of study: Cooling without the use of electricity”, Journal of Physics: Conference Series, 1388 (2019), 012034, 7 pp. | DOI

[17] Marchesse Y., Changenet C., “Forced convective heat transfer over a non-circular slender cylinder”, Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science, 223:2 (2008), 427–437 | DOI

[18] Ivanov D.Yu., “O reshenii ploskikh zadach nestatsionarnoi teploprovodnosti kollokatsionnym metodom granichnykh elementov”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2017, no. 50, 9–29 | DOI

[19] Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy oblasti v sluchae dvumernykh zadach nestatsionarnoi teploprovodnosti s granichnymi usloviyami vtorogo i tretego roda”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2019, no. 57, 5–25 | DOI

[20] Ivanov D.Yu., “Utochnenie kollokatsionnogo metoda granichnykh elementov vblizi granitsy dvumernoi oblasti s pomoschyu poluanaliticheskoi approksimatsii teplovogo potentsiala dvoinogo sloya”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2020, no. 65, 30–52 | DOI

[21] Ivanov D.Yu., “Ekonomichnyi metod vychisleniya operatorov, razreshayuschikh nekotorye zadachi teploprovodnosti v pryamykh tsidindrakh”, Aktualnye problemy gumanitarnykh i estestvennykh nauk, 2014, no. 9, 16–32

[22] Ivanov D.Yu., “Vychislenie operatorov, razreshayuschikh zadachi teploprovodnosti v pryamykh tsilindrakh, s ispolzovaniem polugruppovoi simmetrii”, Izvestiya Moskovskogo gosudarstvennogo tekhnicheskogo universiteta MAMI, 4:4 (22) (2014), 26–38

[23] Budak B.M., Samarskii A.A., Tikhonov A.N., Sbornik zadach po matematicheskoi fizike, Nauka, M., 1979, 685 pp.

[24] Ivanov D.Yu., “Zamknutost summ neogranichennykh operatorov, deistvuyuschikh po raznym peremennym v prostranstvakh kvadratichno summiruemykh funktsii neskolkikh peremennykh”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2017, no. 45, 35–48 | DOI

[25] Lyantse V.E., Storozh O.G., Metody teorii neogranichennykh operatorov, Naukova dumka, Kiev, 1983, 212 pp.

[26] Ivanov D.Yu., “Reshenie dvumernykh kraevykh zadach, sootvetstvuyuschikh nachalno-kraevym zadacham diffuzii na pryamom tsilindre”, Differents. uravneniya, 46:8 (2010), 1094–1103 | DOI

[27] Ivanov D.Yu., Dzerzhinskii R.I., “Reshenie zadach Robena dlya dvumernykh differentsialno-operatornykh uravnenii, opisyvayuschikh teploprovodnost v pryamom tsilindre”, Nauchno-tekhnicheskii vestnik Povolzhya, 2016, no. 1, 15–17

[28] Ivanov D.Yu., “Ustoichivaya razreshimost v prostranstvakh differentsiruemykh funktsii nekotorykh dvumernykh integralnykh uravnenii teploprovodnosti s operatorno-polugruppovym yadrom”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2015, no. 38, 33–45 | DOI

[29] Smirnov V.I., Kurs vysshei matematiki, Ch. 2, v. 4, Nauka, M., 1981, 551 pp.

[30] Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M., Chislennye metody, BINOM. Laboratoriya znanii, M., 2008, 636 pp.