# 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.

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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.
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Section
Mathematical Modeling