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Parsing Abnormal Grain Growth A. Lawrence, J. M. Rickman, M. P. Harmer, Y. Wang Dept. of Materials Science and Engineering Lehigh University A. D. Rollett Dept. of Materials Science and Engineering Carnegie Mellon University
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Characterizing Abnormal Grains Analogy: Rare Events - Insurance losses from one-hundred year flood - Occurrence of forest fires Tools - Extreme-value theory: asymptotic tail distribution - Conditional-tail expectation: moments of special distribution How large are abnormal grains? How many “large” grains must be present? Are abnormal grains elongated relative to “normal” grains? What are the signatures of abnormality?
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Extreme Values of a Distribution How do we characterize the tail of the distribution? Use conditional moments define cutoff
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Microstructures and Bivariate Probability Plots the few grains more grains
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Conditional Tail Expectation: Metrics Generalization average grain size grain-size variance mixed moment
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Other Metrics Exceedance Correlation What is the probability of being in the tail of the distribution? What is the correlation of grain size and aspect ratio for a given microstructure?
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- How can one quantify deviations from “normality”? -Assume that the pdf for “normal” grain growth is given by - Therefore, G and a are uncorrelated. Probability Density Function for Normal Grain Growth -Metric normalization: Compare pdf with p normal. Normalized metrics: i (i=1,2,3,4,5)
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Samples - 33 specialty alumina samples - 11 different compositions - 1000-9000 grains analyzed per microstructure MgO CaO Na 2 O SiO 2 Chemistry Industrial Partner: Almatis, Inc. (Leetsdale, PA) - Powders were spark-plasma sintered. gallery.asiaforest.org c MgO /(c CaO +c SiO2 )
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Microstructures and Bivariate Probability Plots
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Values of Metrics: Microstructural Correlation variance average grain size mixed moment exceedancecorrelation
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Canonical Correlation Analysis (CCA) -Find directions for both input and output variables that are maximally correlated. - Dimensional reduction strategy. Variable Sets Compositions: MgO, CaO, Na 2 O,SiO 2, ratio Temperature Time Microstructural metrics Inputs Outputs
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Procedure Construct covariance matrix - Construct operators - Diagonalize Eigenvalues - measures strength of correlation Eigenvectors - coefficients for canonical variates 1.) 2.)
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Canonical Variates and Loadings Results of Hypothesis Test (Wilk’s Lambda) 1 = 0.92 (p=0.001) Other eigenvalues are not significant. Variates - Linear combination of variables
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Canonical Variate Correlation Canonical Variate, V Canonical Variate, W
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Bubble Plot: Canonical Weights and Loadings Loadings Weights
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Planning Experiments Sensitivity Analysis – omitting input and output variables Processing Variables Metrics
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Conclusions and Ongoing Work -New metrics were developed to quantify abnormality using extreme-value methodologies. -Canonical correlation analysis (CCA) was employed to assess correlations among processing variables and metrics. -CCA can be used for experimental planning. - Additional data sets are needed to test robustness of approach. - More experimental verification is needed.
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