Higher-Order Microstructure Statistics for Next Generation Materials Taxonomy

Tony Fast

University of California Santa Barbara, Materials Engineering

Olga Wodo, Baskar Ganapathysubramanian 

Iowa State University, Mechanical Engineering

Surya R. Kalidindi

Drexel University, Mechanical Engineering

From Materials Selection

μS informatics distill rich spatial and temporal information into tractable, usable, and searchable bi-direction SPP linkages

From Materials Selection to Microstructure (μS) Informatics…

A. 2-pt Correlation Function – Statistical correlation between random points in space/time

B. Chord Length Distribution – length and orientation of chords in a heterogeneous medium

C. Interfacial Surface Distribution - The principal curvatures of surfaces in the μS.

A. B. C.

Effective statistics are contained in μS InformaticsStatistical spatial distributions capture traditional effective statistical measures

C. Kwon, Yongwoo, Morphology and topology of interfaces during coarsening via nonconserved and conserved dynamics, Northwestern, Thesis, 2007.

Benefits of Using n-Point Correlations• Ground Truth• Fit naturally in higher-orderhomogenization and localization theories

The Microstructure is a stochastic processDistributions provide a framework to effectively compare microstructures

Mic

rost

ruct

ure

Aut

ocor

rela

tion

-

- =

=

HT1 HT2 Difference

Extremely Large Dimensional Spaces!

The comparison of μS is dubious due to the lack of origin.

Autcorrelation contains all of the information in its respective μS.

MS informatics benefits from dimensional reductionReducing the number of random variables for feature selection and extraction in discrete materials systems

Improve Empirical Fitting:

Porous Bi-layers in Fuel Cells

Microstructure Taxonomy:

MS Mapping of α-βTitanium

VfKalidindi, S.R., S.R. Niezgoda, and A.A. Salem, Microstructure informatics using higher-order statistics and efficient data-mining protocols. JOM, 2011. 63(4): p. 34-41.

A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, Data-driven Approaches to Establishing Microstructure-property Relationships: Application to Transport through Porous Structures, submitted, 2012.

Principal Component Analysis: Reduced embedding of linearly independent variables that correspond to decreasing levels of variance starting with the highest (Dd)

>6e6 Variables~1e6 Variables

μS Taxonomy of Continuous Material Feature An Application to Organic Blends in Solar Cells

10% FAST PHASE SEPARATION 90% SLOW GRAIN COARSENING

FINAL STRUCTURES

11 Distinct TopologiesMany Topologies

Simulation data provided by Olga Wodo and Baskar Ganapathysubramanian at ISU.

Isosurfaces of atomic fraction1100 Datasets

Use data driven techniques to classify the final topology before the simulation is complete. i.e. Reduce Redundancy, Time Savings

End Goal

Develop Microstructure Taxonomies of the Final Structuresi.e. Build Utilities for Continuous Materials Features

First Goal

First-Order Higher-Order

Discrete

Continuous

μS function of continuous materials featuresInformatics benefit from a generalized higher-order microstructure description

H

hs

hhs vm

1

10,11

hs

H

h

hs mmPrimitive Basis

Function

nsm

nm1nm3

nm5

nm6

nm2

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N

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Local conformation of pixels

ssss ,,~

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μS informatics workflow is a systemMicrostructure Descriptor, Statistics, PCA, etc are isolated modules

Systems analysis allows one to prove the efficacy of methods

HO Descriptors

NP CorrelationsPCA/k-means

Future Work

1100 Datasets

Reduced embedding of final topologiesPCA projection changes with the number of basis functions and the gradient

Hard clustering in the PCA space allows the final topologies to be classified qualitatively

Each point indiciates a 21x21x21 μS where each color is a different topology

Augmented embedding combines gradientsPCA embedding changes when different descriptors are combined

Data-mining with k-means clusteringAutomated topology recognition and quantitative metrics

k-Means Clustering: A data-mining approach that creates partitions based on the means of clusters to automatically classify datapoints.

Sensitivity – metric for accurate classification

Specificity – metric for accurate nonclassification

Classification CasesTP – Correct classificationTN – Correct nonclassificationFP – Incorrect classificationFN – Incorrect nonclassification

Range = 0 to 1

Quantitative Measures of ClusteringSensitivity and specificity analysis of PCA embedding

AF

FG

SGAF

FG

SG

AF – Atomic FractionFG – First GradientSG – Second Gradient

ClockwiseFG,SGAF,SGAF,FGAF,FG,SG

• An automated data-mining technique was successfully developed for 3D systems with continuous μS features.

• A generalized higher-order μS descriptor was developed using the primitive basis.

• Higher-order descriptors prove that higher-order terms play a strong role in developing structure-structure databases.

• This system naturally clusters in PCA, but other DR techniques show improvement.

• μS informatics are necessary to automatically disseminate structure-structure relationships of large collections of multi-dimensional datasets

ConclusionsHigher-Order Microstructure Statistics for Next Generation Materials Taxonomy