# Checkerboard Microstructure

## Introduction - What are 2-Point Spatial Correlations (also called 2-Point Statistics)?

The purpose of this example is to introduce 2-point spatial correlations and how they are computed, using PyMKS.

The example starts with some introductory information about spatial correlations. PyMKS is used to compute both the periodic and non-periodic 2-point spatial correlations (also referred to as 2-point statistics or autocorrelations and crosscorrelations) for a checkerboard microstructure. This is a relatively simple example that allows an easy discussion of how the spatial correlations capture the main features seen in the original microstructure. If you would like more technical details about 2-point statistics please see the theory section.

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In [1]:

import pymks

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

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## 2-Point Statistics for Checkerboard Microstructure

Let's first start with making a microstructure that looks like a 8 x 8 checkerboard. Although this type of microstructure may not resemble a physical system, it provides solutions that give some intuitive understanding of 2-point statistics.

We can create a checkerboard microstructure using `make_checkerboard_microstructure` function from `pymks.datasets`.

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In [2]:

from pymks.datasets import make_checkerboard_microstructure

X = make_checkerboard_microstructure(square_size=21, n_squares=8)

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Now let's take a look at how the microstructure looks.

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In [3]:

from pymks.tools import draw_microstructures

draw_microstructures(X)

print X.shape

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(1, 168, 168)

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## Compute Periodic 2-Point Statistics

Now that we have created a microstructure to work with, we can start computing the 2-point statistics. Let's start by looking at the periodic autocorrelations of the microstructure and then compute the periodic crosscorrelation. This can be done using the `autocorrelate` and `crosscorrelate` functions from `pymks.states`, and using the keyword argument `periodic_axes` to specify the axes that are periodic.

In order to compute 2-pont statistics, we need to select a basis to generate the microstructure function `X_` from the microstructure `X`. Because we only have values of 0 or 1 in our microstructure we will using the `PrimitiveBasis` with `n_states` equal to 2.

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In [4]:

from pymks.stats import autocorrelate
from pymks import PrimitiveBasis

p_basis = PrimitiveBasis(n_states=2)
X_auto = autocorrelate(X, p_basis, periodic_axes=(0, 1))

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We have now computed the autocorrelations.

Let's take a look at them using `draw_autocorrelations` from `pymks.tools`.

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In [5]:

from pymks.tools import draw_autocorrelations

correlations = [('black', 'black'), ('white', 'white')]
draw_autocorrelations(X_auto[0], autocorrelations=correlations)

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Notice that for this checkerboard microstructure, the autocorrelation for these 2 local states in the exact same. We have just computed the periodic autocorrelations for a perfectly periodic microstructure with equal volume fractions. In general this is not the case and the autocorrelations will be different, as we will see later in this example.

As mentioned in the introduction, because we using an indicator basis and the we have eigen microstructure functions (values are either 0 or 1), the (0, 0) vector equals the volume fraction.

Let's double check that both the phases have a volume fraction of 0.5.

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In [6]:

center = (X_auto.shape[1] + 1) / 2
print 'Volume fraction of black phase', X_auto[0, center, center, 0]
print 'Volume fraction of white phase', X_auto[0, center, center, 1]

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Volume fraction of black phase 0.5
Volume fraction of white phase 0.5

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We can compute the cross-correlation of the microstructure function, using the `crosscorrelate` function from `pymks.stats`

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In [7]:

from pymks.stats import crosscorrelate

X_cross = crosscorrelate(X, p_basis, periodic_axes=(0, 1))

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Let's take a look at the cross correlation using `draw_crosscorrelations` from `pymks.tools`.

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In [8]:

from pymks.tools import draw_crosscorrelations

correlations = [('black', 'white')]
draw_crosscorrelations(X_cross[0], crosscorrelations=correlations)

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Notice that the crosscorrelation is the exact opposite of the 2 autocorrelations. The (0, 0) vector has a value of 0. This statistic reflects the probablity of 2 phases having the same location. In our microstructure, this probability is zero, as we have not allowed the two phases (colored black and white) to co-exist in the same spatial voxel.

Let's check that it is zero.

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In [9]:

print 'Center value', X_cross[0, center, center, 0]

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Center value 3.74121770979e-17

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## Compute Non-Periodic 2-Point Statistics

We will now compute the non-periodic 2-point statistics for our microstructure. This time, rather than using the `autocorrelate` and `crosscorrelate` functions, we will use the `correlate` function from `pymks.stats`. The `correlate` function computes all of the autocorrelations and crosscorrelations at the same time. We will compute the non-periodic statistics by omitting the keyword argument `periodic_axes`.

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In [10]:

from pymks.stats import correlate

X_corr = correlate(X, p_basis)

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All or some of the correlations can be viewed, using the `draw_correlations` function from `pymks.tools`. In this example we will look at all of them.

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In [11]:

from pymks.tools import draw_correlations

correlations = [('black', 'black'), ('white', 'white'), ('black', 'white')]
draw_correlations(X_corr[0].real, correlations=correlations)

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Notice that the maximum values for the autocorrelations are higher than 0.5. We can still show that the centers or the (0, 0) vectors are still equal to the volume fractions.

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In [12]:

print 'Volume fraction of black phase', X_corr[0, center, center, 0]
print 'Volume fraction of white phase', X_corr[0, center, center, 1]

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Volume fraction of black phase 0.5
Volume fraction of white phase 0.5

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The non-periodic statistics are different from the periodic 2-point statistics along the diagonal vectors, but in both cases the probability of (0, 0) vector is still the volume fraction.

## References

[1] S.R. Niezgoda, D.T. Fullwood, S.R. Kalidindi, Delineation of the Space of 2-Point Correlations in a Composite Material System, Acta Materialia, 56, 18, 2008, 5285–5292 doi:10.1016/j.actamat.2008.07.005

[2] S.R. Niezgoda, D.M. Turner, D.T. Fullwood, S.R. Kalidindi, Optimized Structure Based Representative Volume Element Sets Reflecting the Ensemble-Averaged 2-Point Statistics, 58, 13, 2010, 4432–4445 doi:10.1016/j.actamat.2010.04.041

[3] D.T. Fullwood, S.R. Kalidindi, and B.L. Adams, Second - Order Microstructure Sensitive Design Using 2-Point Spatial Correlations, Chapter 12 in Electron Backscatter Diffraction in Materials Science , 2nd Edition , Eds. A. Schwartz, M. Kumar, B. Adams, D. Field, Springer, NY, 2009.