Mathematical Modeling of Temperature Regime of Soils of Foundation on Permafrost
Authors: Vasilyev V.I., Vasilyeva M.V., Sirditov I.K., Stepanov S.P., Tseeva A.N. | Published: 14.02.2017 |
Published in issue: #1(70)/2017 | |
DOI: 10.18698/1812-3368-2017-1-142-159 | |
Category: Informatics, Computer Engineering and Control | Chapter: Mathematical Modelling. Numerical Methods, and Software Systems | |
Keywords: numerical simulation, Stefan problem, finite-element method, permafrost, high-perfomance computing systems, pile foundations, software |
In the article we consider a mathematical model describing the heat transfer processes in the frozen and thawed soils. We provide a mathematical formulation of the problem with the appropriate initial and boundary conditions. Numerical implementation is based on the finite-element method, which allows us to take into account the complication caused by difficulties with geometry, since piles are small comparing with whole domain. In this paper we include the comparison of two-dimensional and threedimensional models of heat transfer calculation in a soil, taking into account the installation of piles and the seasonal fluctuations of ambient temperature. We present the results of numerical simulation of soil temperature on the example of engineering solutions containing several piles installed in the soil with a layered structure. We discuss the elements of the developed applied software for the predicting the temperature regime of soils on permafrost. Numerical simulations of the problem in three-dimensional case on the computational cluster of NEFU “Arian Kuzmin” are provided.
References
[1] Samarskii A.A., Vabishchevich P.N. Computational heat transfer. Vol. 1. Mathematical Modelling. Wiley, 1995. 418 p.
[2] Vasilyev V.I., Maksimov A.M., Petrov E.E., Tsypkin G.G. Teplomassoperenos v promerzayushchikh i protaivayushchikh gruntakh [Heat and mass transfer in freezing and melting soils]. Moscow, Nauka Publ., 1996. 224 p.
[3] Vasilyev V.I., Popov V.V. Numerical solution of the soil freezing problem. Math. Models Comput. Simul., 2009, vol. 1, iss. 4, pp. 419-427. DOI: 10.1134/S2070048209040012
[4] Pavlov A.V. Teplofizika landshaftov [Thermophysics of landscapes]. Novosibirsk, Nauka Publ., 1979. 284 p.
[5] Samarskii A.A. The theory of difference schemes. N.Y.-Basel, Marcel Dekker, Inc., 2001. 761 p.
[6] Vasilyeva M.V., Pavlova N.V. Finite element implementation of the task of filtering grounds freezing. Matematicheskie zametki JaGU [Math. Notes of Yakutsk State Univ.], 2013, vol. 20, no. 1, pp. 195-205 (in Russ.).
[7] Logg Anders, Mardal Kent-Andre, Wells Garth N. Automated solution of differential equations by the finite element method. The FEniCS Book, 2011. 732 p.
[8] Krylov D.A., Sidnyaev N.I., Fedotov A.A. Mathematical modelling of temperature distribution. Matem. Mod., 2013, vol. 25, no. 7, pp. 3-27 (in Russ.).
[9] Gornov V.F., Stepanov S.P., Vasilyeva M.V., Vasilyev V.I. Mathematical modeling of heat transfer problems in the permafrost. AIP Conference Proceedings, 2014, vol. 1629, pp. 424-431.
[10] Vabishhevich P.N., Vasilyeva M.V., Pavlova N.V. Numerical simulation of thermal stabilization of filter soils. Math. Models Comput. Simul., 2015, vol. 7, iss. 2, pp. 154-164. DOI: 10.1134/S2070048215020106
[11] Vabishchevich P.N., Vasilyeva M.V., Gornov V.F., Pavlova N.V. Mathematical modeling of the artificial freezing of soils. Vychislitel’nye tekhnologii [Computational Technologies], 2014, vol. 19, no. 4, pp. 19-31 (in Russ.).
[12] Vabishchevich P.N., Varlamov S.P., Vasilyev V.I., Vasilyeva M.V., Stepanov S.P. Mathematical modeling of the thermal regime of a railway line in conditions of cryolithozone. Vestnik SVFU [Vestnik of North-Eastern Federal Univ.], 2013, vol. 10, no. 5, pp. 5-11 (in Russ.).
[13] Saad Y. Iterative methods for sparse linear systems. Society for Industrial Mathematics. 2003. 528 p.
[14] Voevodin V.V., Voevodin Vl.V. Parallel’nye vychisleniya [Parallel computing]. St. Petersburg, BKhV-Petersburg Publ., 2002. 608 p.
[15] Antonov M.Yu., et al. Vychislitel’nye tekhnologii. Professional’nyy uroven’ [Computational technologies. Professional level]. Yakutsk, Izdatel’skiy dom SVFU Publ., 2014. 308 p.