Robert Derr

Statistical Modeling of Microstructure with Applications
to Effective Property Computation in Materials Science

(Under the direction of Dr. Chuanshu Ji)

We propose a class of models for microstructure in materials science, and conduct a statistical modeling process: perform image processing of the microstructure to summarize the data, perform inference and parameter estimation based on this summary data, use spatial birth-and-death processes to create Markov chain Monte Carlo simulations of the structures, perform post-modeling diagnostics and evaluate the goodness of fit via feature extraction. The proposed class of model is a hard-core/soft-shell elliptical point process--a special case of Markov point processes on geometric shapes, originally proposed by Ripley and Kelly (1977) and Baddeley and Moller (1989).

This model may be utilized for the computation of certain properties, such as conductivity, of heterogeneous materials following the original work of Brown (1954) and Torquato (1985). Taking their analytical framework, one difficulty along this line is the approximation of a family of k-point probability functions (the probability that k points in a given configuration all lie within one phase of the material); these functions play a critical role in the computations. Using the elliptical point process model and Monte Carlo procedures, we study the characteristics of such functions, propose a model of the k-point probabilities based on the 2-point probability function, and compare the two approaches.