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Current Projects
Materials Science Research: Computing phase diagrams for ternary alloys using low-discrepancy sequences
Mentor: Dr. Maria Emelianenko
Dr. Emelianenko and I are working on the efficient and accurate computation of phase diagrams for multicomponent systems. Phase diagrams can be determined by examining the Gibbs free energy functions associated with some material and the various phases of matter it can assume. In multicomponent systems, these free energy functions get complicated, and the free energy functions describing individual phases of matter must be considered simultaneously, which sets up complex scenarios. The phase diagram reveals the most thermodynamically advantageous states of matter that the system will assume for a given temperature and composition. Determining these configurations involves finding minimal energy envelopes among that form within each individual free energy function, as well as those that form among the multiple free energies considered simultaneously, and this is the computationally difficult part.
The domain of our problem (for a ternary system) is the Gibbs triangle, and the free energy can be thought of as describing a surface above the domain. In general these surfaces can have multiple wells, or local minima, and each of these low-lying regions must be identified in order to accurately determine the phase diagram. (Multiple minima imply the presence of a miscibility gap, and when I understand more of the physics behind this I will update this description.) Computationally, this is a difficult problem. One approach would be to continue to refine some discretization of the domain and compute the free energy function at the grid points. But this method is supremely inefficient -- as the resolution grows linearly, the computational effort grows as a polynomial. In other words, in order to achieve just a bit more accuracy, the problem will take much more effort. The method we are working on uses low-discrepancy sequences to sample the function on the Gibbs triangle, and employs some adaptive searching methods in order to avoid re-computing on a finer discretization over the entire domain.
Some of the tools we are using are k-means clustering of data points and Niederreiter and Peart's localization of search algorithm. I am scheduled to present our progress at the SIAM MS10 conference in May 2010. Check back for updates.
Code
Throughout the day I find myself writing code to do one thing or another. Recently I have been writing a good deal of object oriented Matlab code, which isCoursework
Fall 2008Spring 2009
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