Numerical homogenization for heat transfer problems in the permafrost zone

  • Alekseev Valentin N., alekseev.valen@mail.ru International Research Laboratory "Multiscale Model Reduction", Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677980, Russia
  • Tyrylgin Aleksei A., koc9tk@mail.ru International Research Laboratory "Multiscale Model Reduction", Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677980, Russia
  • Vasilyeva Maria V., vasilyevadotmdotv@gmail.com Department of Computational Technology, Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677980, Russia; Institute for Scientific Computation, Texas A&M University, College Station, TX 77843-3368
  • Vasilyev Vasiliy I., vasvasil@mail.ru Department of Computational Technology, Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677980, Russia
Keywords: mathematical modeling, heat transfer, phase transition, the Stefan problem, numerical homogenization, the finite element method, FEniCS

Abstract

The work considers heat transfer problems taking into account phase transitions of moisture in the soil. The mathematical model of heat transfer processes with phase transition is described using the classical Stefan model and is a nonlinear parabolic equation. To solve the problem, a numerical homogenization method is proposed for the nonlinear problem using the effective thermal conductivity coefficient for thawed and frozen zones. The calculation of the effective thermal conductivity tensor is carried out in local domains (coarse mesh cells) and is used to construct the approximation on a coarse mesh by the finite element method. Numerical implementation was carried out using FEniCS computational library for finite element approximation. Numerical results are presented for the model problem in two-dimensional and three-dimensional formulations.

References


[1]
Vabishchevich P. V., Varlamov S. P., Vasilyev V. I., Vasilyeva M. V., and Stepanov S. P., “Numerical simulation of the temperature field of the permafrost soil base of the railway [in Russian],” Mat. Modelir., 28, No. 10, 110–124 (2016).

[2]
Vabishchevich P. V., Varlamov S. P., Vasilyev V. I., Vasilyeva M. V., and Stepanov S. P., “Mathematical modeling of the thermal regime of the railway in the conditions of cryolithozone [in Russian],” Vestn. Severo-Vostoch. Feder. Univ. im. M. K. Ammosova, 10, No. 5, 5–11 (2013).

[3]
Vasilyev V. I., Danilov Y. G., Eremeev I. S., Popov V. V., Tsypkin G. G., Yuyzhui S., and Yandong Z., “Comparison of mathematical models of heat and mass transfer in soils [in Russian],” Vestn. Severo-Vostoch. Feder. Univ. im. M. K. Ammosova, 10, No. 4, 5–10 (2013).

[4]
Vasilyeva M. V. and Pavlova N. V., “Finite-element implementation of the problem of freezing filter soils [in Russian],” Mat. Zamet. SVFU, 20, No. 1, 195–205 (2013).

[5]
Pavlov A. V., Perlstein G. Z., and Tipenko G. S., “Actual aspects of modeling and prediction of the thermal state of the permafrost zone in a changing climate [in Russian],” Kriosfera Zemli, 14, No. 1, 3–12 (2010).

[6]
Vabishchevich P. N., Vasilyeva M. V., and Pavlova N. V., “Numerical simulation of thermal stabilization of filtering soils [in Russian],” Mat. Modelir., 26, No. 9, 111–125 (2013).

[7]
Vabishchevich P. N. and Iliev O. P., “Chislennoe reshenie soprjazhennyh zadach teplo- i massoperenosa s uchetom fazovogo perehoda [in Russian],” Differ. Uravn., 23, No. 7, 1127–1132 (1987).

[8]
Stepanov S. P., Sirditov I. K., Vabishchevich P. N., Vasilyeva M. V., Vasilyev V. I., and Tseeva A. N., “Numerical simulation of heat transfer of the pile foundations with permafrost,” in: Int. Conf. Numerical Analysis and Its Applications, pp. 625–632 (2016).

[9]
Vasilyeva M. V. and Stalnov D. A.,, “Numerical averaging for the heat conduction problem in heterogeneus and perforated media [in Russian],” Vestn. Severo-Vostoch. Feder. Univ. im. M. K. Ammosova, 2, No. 58, 49–59 (2017).

[10]
Talonov A. and Vasilyeva M., “On numerical homogenization of shale gas transport,” J. Comput. Appl. Math., 301, 44–52 (2016).

[11]
Savatorova V. L., Talonov A. V., and Vlasov A. N. , “Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities,” ZAMM, J. Appl. Math. Mech., Mechanik, 93, No. 8, 575–596 (2013).

[12]
Tyrylgin A., Spiridonov D., and Vasilyeva M., “Numerical homogenization for poroelasticity problem in heterogeneous media,” in: J. Phys. Conf. Ser. IOP Publ., 1158, No. 4, 042030 (2019).

[13]
Gavrilieva U., Alekseev V., and Vasilyeva M., “Numerical homogenization for wave propagation in fractured media,” in: AIP Conf. Proc., 2025, No. 1, 100002, AIP Publ. (2018).

[14]
Alekseev V., Gavrilieva U., Spiridonov D., Tyrylgin A., and Vasilyeva M., “Numerical simulation of the transport and flow problems in perforated domains using generalized multiscale finite element method,” in: AIP Conf. Proc., 2025, No. 1, 100001, AIP Publ. (2018).

[15]
Stepanov S., Vasilyeva M., Vasil’ev V., “Generalized multiscale discontinuous Galerkin method for solving the heat problem with phase change,” J. Comput. Appl. Math., 340, 645–652 (2018).

[16]
Chung E., Efendiev Y., Leung W., and Ren J., “Multiscale simulations for coupled flow and transport using the generalized multiscale finite element method,” Computation, 3, No. 4, 670–686 (2015).

[17]
Alekseev V., Tyrylgin A., and Vasilyeva M., “Generalized multiscale Finite Element Method for elasticity problem in fractured media,” in: Int. Conf. Finite Difference Methods, pp. 137–144, Springer, Cham (2015).

[18]
Geuzaine C. and Remacle J. F., “Gmsh: A 3-D finite element mesh generator with builtin pre-and post-processing facilities,” Int. J. Numer. Methods Eng., 79, No. 11, 1309–1331 (2015).

[19]
Logg A., Mardal K.-A., and Wells G., eds., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS book, vol. 84, Springer (2012).
How to Cite
Alekseev, V., Tyrylgin, A., Vasilyeva, M. and Vasilyev, V. (2020) “Numerical homogenization for heat transfer problems in the permafrost zone”, Mathematical notes of NEFU, 27(2), pp. 77-92. doi: https://doi.org/10.25587/SVFU.2020.47.81.005.
Section
Mathematical Modeling