Downloads & Free Reading Options - Results

Under support from this grant, we performed work on simulation and design for a number of important topics. These included level set modeling of dendritic solidification, regularized Wulff flows, singularities for flow in porous media, evaluation of American options, epitaxial growth and strain, and electrodeposition. Key results include the following: 1) We developed a new reduced order model for epitaxial growth. The average coverage, the average island size and the average inter-island distance at each layer of the system. The resulting equations are a system of 3 eta ODEs for a epitaxial film of thickness eta layers. This system displays good agreement with the oscillations observed in RHEED measurements during MBE growth, including the variations in the envelope of the oscillations. We expect this model to be useful both for basic understanding and as a tool for control methods. 2) We presented a level set approach for the modeling of dendritic solidification. These simulations used a new second order accurate symmetric discretization of the Poisson equation. Numerical results indicated that this method can be used successfully on complex interfacial shapes and can simulate many of the physical features of dendritic solidification. We applied this algorithm to the simulation of the dendritic crystallization of a pure melt and find that the dendrite tip velocity and tip shapes are in excellent agreement with solvability theory. Numerical results were presented in both two and three spatial dimensions. 3) We proposed a method of regularizing the backwards parabolic partial differential equations that arise from using gradient descent to minimize surface energy integrals within a level set framework in two and three dimensions. The proposed regularization energy is a functional of the mean curvature of the surface. Our method used a local level set technique to evolve the resulting fourth order PDEs in time.