Stochastic Modeling and Rare Event Simulation for Gibbs Distributions with Applications in Materials Engineering

Shruthi S Kubatur, Purdue University

Abstract

Many important physical processes in fields such as materials science, ecology, structural biology, and clinical pathology involve the study of microscopic structures—from formation and propagation to steady-state behavior. Direct observation of these phenomena is often very slow and expensive, creating an enormous need for accurate computer simulation of the underlying processes. Often, certain events of low probability that occur in material systems have a considerable impact on system design. Though rare-event simulation has been a well-researched problem in areas such as financial risk assessment and communication systems, modeling and simulation of rare events in materials systems remain severely under-explored. In this work, we propose importance sampling algorithms based on the theory of large deviations, along with a Markov chain Monte Carlo (MCMC) framework to enable simulation of low-probability events that are critical to engineering material systems. Specifically, we explore two phenomena in materials science—abnormal grain growth in polycrystalline materials, and overlapping precipitates in an important class of Ni-based super-alloys. The micro-structure of certain polycrystalline materials consists of grains that have different orientations associated with them. These grains evolve over time and this phenomenon is called grain growth. However, an event of interest which occurs with low probability involves a single grain that grows abnormally large at the expense of other grains. Though Gibbs distribution-based models exist for grain growth, occurrence of abnormal grain growth under such models is still rare enough that we still need to draw many samples before an abnormal growth manifests. We propose an importance sampling distribution from which to draw samples to simulate abnormal grain growth, instead of the conventional Gibbs distribution. We show that the proposed importance sampling distribution leads to an asymptotically efficient rare- event probability estimator. Next, we present an algorithm for simulation of Ni-based super-alloy precipitates modeled by marked point processes using a Gibbs distribution. Further, we pro- pose an importance sampling distribution—inspired by the mathematical framework adopted in the abnormal grain growth simulation algorithm – to simulate rarely- occurring overlapping precipitates in NiCrAl super-alloys. We present simulation results for both applications, propose a maximum likelihood method for estimating parameters of simulation, and present a method for model validation using anomaly detection.

Degree

Ph.D.

Advisors

Comer, Purdue University.

Subject Area

Electrical engineering

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