Finite Element Simulation of Thermo-Mechanical Model with Phase Change

Abstract

In this work, we consider a mathematical model and finite element implementation of heat transfer and mechanics of soils with phase change. We present the construction of the simplified mathematical model based on the definition of water and ice fraction volumes as functions of temperature. In the presented mathematical model, the soil deformations occur due to the porosity growth followed by the difference between ice and water density. We consider a finite element discretization of the presented thermoelastic model with implicit time approximation. Implementation of the presented basic mathematical model is performed using FEniCS finite element library and openly available to download. The results of the numerical investigation are presented for the two-dimensional and three-dimensional model problems for two test cases in three different geometries. We consider algorithms with linearization from the previous time layer (one Picard iteration) and the Picard iterative method. Computational time is presented with the total number of nonlinear iterations. A numerical investigation with results of the convergence of the nonlinear iteration is presented for different time step sizes, where we calculate relative errors for temperature and displacements between current solution and reference solution with the largest number of the time layers. Numerical results illustrate the influence of the porosity change due to the phase-change of pore water into ice on the deformation of the soils. We observed a good numerical convergence of the presented implementation with the small number of nonlinear iterations, that depends on time step size.

On a question of non-constant thermal diffusivity of soils

“…Various soils have similar dependence of thermal conductivity on bulk density. According to [30], it is expressed by the following equation: The common formulae for the thermal conductivity for a composite material [19,43] for wet soil with mineral and organic solid particles, liquid, and gaseous phases will take a form:…”