Numerical Simulation of Liquid-Solid, Solid-Liquid Phase Change Using Finite Element Method in h,p,k Framework with Space-Time Variationally Consistent Integral Forms

Abstract

This thesis presents development of mathematical models for liquid-solid phase change phenomena using Lagrangian description with continuous and differentiable smooth interface (transition region) between the solid and the liquid phases in which specific heat, thermal conductivity, and latent heat of fusion are a function of temperature. The width of the interface region can be as small or as large as desired in specific applications. The mathematical models presented in the thesis assume homogeneous and isotropic medium, zero velocity field (no flow) with free boundaries i.e. stress free domain. With these assumptions the mathematical model reduces to the first law of thermodynamics i.e. energy equation. The mathematical models presented here are neither labeled as enthalpy models or others, instead these are based on a simple statement of the first law of thermodynamics using specific total energy and heat vector augmented by the constitutive equation for heat vector i.e. Fourier heat conduction law and the statement of total specific energy incorporating the physics of phase change in the smooth interface region between solid and liquid phases. This results in a time dependent non-linear convection diffusion in temperature in which physics of interface initiation and propagation is intrinsic and thus avoids front tracking methods. This can also be cast as a system of first order PDEs using auxiliary variables and auxiliary equations if so desired due to the use of specific methods of approximation as done in the present work. The numerical solutions of the initial value problems resulting from the mathematical models are obtained using space-time least squares finite element process based on minimization of the residual functional. This results in space-time variationally consistent integral forms that yield symmetric algebraic systems with positive definite coefficient matrices that ensure unconditionally stable computations during the entire evolution. The local approximations for the space-time finite elements are considered in h,p,k framework which permits higher degree as well as higher order local approximations in space and time. Computations of the evolution are performed using a space-time strip or slab corresponding to an increment of time with time marching procedure. 1D numerical studies are presented and the results are compared with sharp interface and phase field methods. Numerical studies also presented for 1D and 2D model problems in which initiation as well as propagation of the interface is demonstrated. These studies cannot be performed using sharp interface and phase field models. The significant aspects of the present work are: (i) the smooth interface permits desired physics and avoids singular fronts that are non physical (ii) the mathematical model resulting from the present approach is a non-linear diffusion equation, hence intrinsically containing the ability to initiate as well as locate the front during evolution and hence no special front tracking methods are needed. (iii) This methodology permits initiation of the interface i.e. it permits initiation of the phase change phenomena. This is not possible in sharp interface and phase field methods. (iv) The computational infrastructure used ensures stable computations and high accuracy of evolution for each time step and hence time accurate evolutions are possible.