Abstract
This thesis presents an efficient numerical solution technique for the computer simulation of microstructural evolutions of materials. The key idea is to use a multi-resolution Wavelet-Galerkin technique to solve the governing equations for the phase field model. To our knowledge, this is the first time that the wavelet transform and the phase field model are brought together. The major advantage of the Wavelet-Galerkin method over traditional numerical methods is its ability to automatically focus on sharp changes (regions of high gradient) of the solution. Using the Wavelet-Galerkin scheme the numerical calculations can be concentrated on the regions of high gradient, while little or no calculation is needed over regions of low gradient. Although a finite element scheme using adaptive meshing can also focus the calculation on the regions of high gradient, the Wavelet-Galerkin scheme is far more easier to implement numerically. The Wavelet-Galerkin scheme suits the phase field model ideally because the phase field model uses the high gradient of the field variables to represent the interfaces inside a material. The multi-resolution wavelet transform (based on Daubechies wavelets) has been implemented for one, two and three dimensions respectively. Data compression was investigated using a set of controlled cases. It was found that only 10% of the original data are sufficient enough to re-construct a 3D image to a good accuracy. A multi-resolution Wavelet-Galerkin solution technique was constructed and implemented for a one dimensional phase field model. The one-dimensional phase field model was represented by the classical diffusion equation plus an extra term such that at the steady state two very localised regions of high gradient separate from each other at a constant speed. This model is a onedimensional analogy of grain-growth. The construction of the multi-resolution Wavelet-Galerkin solution includes calculating the “connection-coefficients” for the diffusion term of the equation and the treatment of the extra term in the wavelet transform. The former is very tedious and time consuming, but once the connection coefficients are calculated, they can be used straightforwardly by any other people. The Wavelet-Galerkin solutions in one-dimensional case were verified by comparing them with the finite difference solutions. Very good agreement between the two results was obtained. Data compression was investigated in the multi-resolution Wavelet-Galerkin solution. A Wavelet-Galerkin scheme using -1 level of resolution was developed and implemented for three-dimensional phase field model. The 3D Wavelet-Galerkin scheme was verified by comparing the Wavelet-Galerkin solution with an analytical solution for a simple case of a spherical grain embedded in an infinitively large matrix. The 3D Wavelet-Galerkin scheme was then used to analyse several 3D cases of grain-growth. Through comparing the CPU times for the solution to reach a common physical time using the Wavelet-Galerkin scheme and the finite difference method respectively, the computation efficiency of Wavelet-Galerkin scheme using -1 level of resolution for three-dimensional grain growth problems was investigated. It was found that the Wavelet-Galerkin scheme is an order of magnitude more efficient than the finite difference method in the three-dimensional problems. Two papers have been submitted based on the results of this PhD project: 1. Wavelet-Galerkin Solution to 1D Phase Field Equations for Microstructural Evolution of Materials 2. Wavelet-Galerkin Solution to 3D Phase Field Equations for Microstructural Evolution of Materials.