Two phase alloys are quite important in materials science and metallurgy. Some common examples include nickel-aluminum system, iron-carbon... Show moreTwo phase alloys are quite important in materials science and metallurgy. Some common examples include nickel-aluminum system, iron-carbon system etc. The most important macroscopic properties of these alloys depend on size, orientation and concentration of the second-phase precipitates. It is necessary to understand the details of formation, growth and equilibrium conditions of these micro-structures for better material production. In this dissertation we investigate the growth of the precipitates within the matrix using a sharp interface approach. We consider the effects of elastic fields on the evolution of the precipitates. The elastic fields can either be applied at the far field or can simply arise as a result of crystallographic difference between matrix and precipitate phase. The precipitates exhibit complicated morphology because of the Mullins-Sekerka instability. Our investigation is based on both analytical and numerical techniques. We use linear analysis to understand the qualitative behavior of the problem, at least for short time. To simulate the long time dynamics of the problem and to understand the effects of nonlinearity, we use highly accurate boundary integral methods. Our main contribution in this thesis is threefold. First, starting from linear analysis, we focus on the conditions under which stable growth, in presence of elastic field, is possible for a single precipitate. Finding such conditions are important in material production and simple conditions like constant material flux and constant elastic fields produce precipitates with complicated shapes. Second, we propose a space-time rescaling of the original boundary integral equations of the problem. The rescaling enables us to accurately simulate very long time behavior of the system comprising of multiple precipitates growing under different mass flux and elasticity. It also helps us to understand the long time interaction of precipitates. Third, we xiii implement an adaptive treecode to reduce the computational complexity of the iterative solver from O(N2) to O(N logN) where N is the dimension of the discrete problem. The efficiency of the treecode is demonstrated by performing simulations. Also a parallelization strategy for the treecode is discussed. The speed-up from the parallelization is demonstrated using moderate number of cores. xiv PH.D in Applied Mathematics, December 2012 Show less
