Multiscale model reduction for a thermoelastic model with phase change using a generalized multiscale finite-element method

The development of the cryolithozone requires building and numerically implementing mathematical models of multiphysics thermoelastic processes involving with first-order phase transitions and occurring in the foundations of engineering structures and buildings. Numerical implementation of such models is associated with computational difficulties due to various types of heterogeneities in applied problems and the nonlinearity of governing equations, which require very fine grids, increasing computational costs. We develop a numerical method for solving a thermoelasticity problem with phase transitions based on the generalized multiscale finite-element method (GMsFEM). The main idea of the GMsFEM is to construct multiscale basis functions that take the medium heterogeneities into account. The approximation on a fine grid is carried out using the finite-element method with standard linear basis functions. To verify the accuracy of the proposed multiscale method, we solve two- and three-dimensional problems in heterogeneous media. Numerical results show that the multiscale method can provide a good approximation to the solution of the thermoelasticity problem with a phase transition on a fine grid with a significant reduction in the dimensionality of the discrete problem.