Presentation given at 49th Annual Technical Meeting for the Society of Engineering Sciences in the Materials - Processing, Microstructure, Performance Relations Symposia on October 12, 2012 at Georgia Tech in Atlanta, GA.

1. 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

2. From Materials Selection to Microstructure (μS) Informatics…
μS informatics distill rich spatial and temporal information into tractable, usable,
and searchable bi-direction SPP linkages

3. Effective statistics are contained in μS Informatics
Statistical spatial distributions capture traditional effective statistical measures
A.
B. C.
Benefits of Using n-Point Correlations
• Ground Truth
• Fit naturally in higher-order
homogenization and localization theories
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.
C. Kwon, Yongwoo, Morphology and topology of interfaces during coarsening via nonconserved and conserved dynamics,
Northwestern, Thesis, 2007.

4. The Microstructure is a stochastic process
Distributions provide a framework to effectively compare microstructures
HT1 HT2 Difference
Microstructure
- =
The comparison of μS is dubious due to the lack of origin.
Autocorrelation
- =
Autcorrelation contains all of the information in its respective μS.
Extremely Large Dimensional Spaces!

5. MS informatics benefits from dimensional reduction
Reducing the number of random variables for feature selection and extraction
in discrete materials systems
Principal Component Analysis: Reduced embedding of linearly independent variables
that correspond to decreasing levels of variance starting with the highest (Dd)
Improve Empirical Fitting: ~1e6 Variables Microstructure Taxonomy: >6e6 Variables
Porous Bi-layers in Fuel Cells MS Mapping of α-βTitanium
Vf
A. Çeçen, T. Fast, E. C. Kumbur, and S. R. Kalidindi, Data-driven Kalidindi, S.R., S.R. Niezgoda, and A.A. Salem, Microstructure
Approaches to Establishing Microstructure-property Relationships: informatics using higher-order statistics and efficient data-mining
Application to Transport through Porous Structures, submitted, 2012. protocols. JOM, 2011. 63(4): p. 34-41.

6. μS Taxonomy of Continuous Material Feature
An Application to Organic Blends in Solar Cells
Many Topologies 11 Distinct Topologies
10% FAST PHASE SEPARATION 90% SLOW GRAIN COARSENING
FINAL STRUCTURES
Use data driven techniques to classify the final topology before the
Isosurfaces of atomic fraction
End Goal
simulation is complete. i.e. Reduce Redundancy, Time Savings
1100 Datasets
Develop Microstructure Taxonomies of the Final Structures
First Goal
i.e. Build Utilities for Continuous Materials Features
Simulation data provided by Olga Wodo and Baskar Ganapathysubramanian at ISU.

7. μS function of continuous materials features
Informatics benefit from a generalized higher-order microstructure description
H H
Primitive Basis h h h h
m v
s s
ms 1, 0 ms 1
Function
h 1 h 1
First-Order Higher-Order m6
n
n n n
m1 ms m3
Local conformation n
m2
0 white / solid
Discrete
h s of pixels m5
n
m s
1 s
black / pore ~
~h
ms
h h
m s 0 m s 1 t1  m s N t N
h
~
s s
,  s , s
Continuous
 f s
h h2
ms 1 f s
ms
~
~
m
h
m m m
h0 h1 h2
s s s s

8. μS informatics workflow is a system
Microstructure Descriptor, Statistics, PCA, etc are isolated modules
1100 Datasets
HO Descriptors
Future Work
PCA/k-means NP Correlations
Systems analysis allows one to prove the efficacy of methods

9. Reduced embedding of final topologies
PCA projection changes with the number of basis functions and the gradient
Each point indiciates a 21x21x21 μS where each color is a different topology
Hard clustering in the PCA space allows the final topologies to be classified qualitatively

10. Augmented embedding combines gradients
PCA embedding changes when different descriptors are combined

11. Data-mining with k-means clustering
Automated 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.
Classification Cases
TP – Correct classification
TN – Correct nonclassification
FP – Incorrect classification
FN – Incorrect nonclassification
Sensitivity – metric for accurate classification
Specificity – metric for accurate nonclassification
Range = 0 to 1

12. Quantitative Measures of Clustering
Sensitivity and specificity analysis of PCA embedding
AF – Atomic Fraction FG FG
FG – First Gradient AF
SG – Second Gradient AF SG
SG
Clockwise
FG,SG
AF,SG
AF,FG
AF,FG,SG

13. Conclusions
Higher-Order Microstructure Statistics for Next Generation Materials Taxonomy
• 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