In this short introduction, we will demonstrate the functionality of PyMKS to compute 2-point statistics in order to objectively quantify microstructures, predict effective properties using homogenization and predict local properties using localization. If you would like more technical details amount any of these methods please see the theory section.

```
In [1]:
```%matplotlib inline
%load_ext autoreload
%autoreload 2
import numpy as np
import matplotlib.pyplot as plt

```
In [2]:
```from pymks.datasets import make_microstructure
X_1 = make_microstructure(n_samples=1, grain_size=(25, 25))
X_2 = make_microstructure(n_samples=1, grain_size=(15, 95))
X = np.concatenate((X_1, X_2))

`X`

is used to represent microstructures. Now that we have made the two microstructures, lets take a look at them.

```
In [3]:
```from pymks.tools import draw_microstructures
draw_microstructures(X)

```
```

`correlate`

function from `pymks.stats`

. This function computes all of the autocorrelations and cross-correlation(s) for a microstructure. Before we compute the 2-point statistics, we will discretize them using the `PrimitiveBasis`

function.

```
In [4]:
```from pymks import PrimitiveBasis
from pymks.stats import correlate
prim_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
X_ = prim_basis.discretize(X)
X_corr = correlate(X_, periodic_axes=[0, 1])

```
In [5]:
```from pymks.tools import draw_correlations
print X_corr[0].shape
draw_correlations(X_corr[0])

```
```

```
In [6]:
```draw_correlations(X_corr[1])

```
```

In this section of the intro, we are going to predict the effective stiffness for two phase microstructures using the `MKSHomogenizationModel`

, but we could have chosen any other effective material property.

First we need to make some microstructures and their effective stress values to fit our model. Let's create 200 random instances 3 different types of microstructures, totaling to 600 microstructures.

```
In [7]:
```from pymks.datasets import make_elastic_stress_random
grain_size = [(47, 6), (4, 49), (14, 14)]
n_samples = [200, 200, 200]
X_train, y_train = make_elastic_stress_random(n_samples=n_samples, size=(51, 51),
grain_size=grain_size, seed=0)

`X_train`

is our microstructures. Throughout PyMKS `y`

is used as either the prpoerty or the field we would like to predict. In this case `y_train`

is the effective stress values for `X_train`

. Let's look at one of each of the three different types of microstructures.

```
In [8]:
```draw_microstructures(X_train[::200])

```
```

`MKSHomogenizationModel`

uses 2-point statistics, so we need provide a discretization method for the microstructures by providing a basis function. We will also specify which correlations we want.

```
In [9]:
```from pymks import MKSHomogenizationModel
prim_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
homogenize_model = MKSHomogenizationModel(basis=prim_basis,
correlations=[(0, 0), (1, 1), (0, 1)])

Let's fit our model with the data we created.

```
In [10]:
```homogenize_model.fit(X_train, y_train, periodic_axes=[0, 1])

Now let's make some new data to see how good our model is.

```
In [11]:
```n_samples = [10, 10, 10]
X_test, y_test = make_elastic_stress_random(n_samples=n_samples, size=(51, 51),
grain_size=grain_size, seed=100)

We will try and predict the effective stress of our `X_test`

microstructures.

```
In [12]:
```y_pred = homogenize_model.predict(X_test, periodic_axes=[0, 1])

`MKSHomogenizationModel`

generates low dimensional representations of microstructures and regression methods to predict effective properties. Take a look at the low dimensional representations.

```
In [14]:
```from pymks.tools import draw_components
draw_components([homogenize_model.reduced_fit_data, homogenize_model.reduced_predict_data],
['Training Data', 'Testing Data'])

```
```

Now let's look at a goodness of fit plot for our `MKSHomogenizationModel`

.

```
In [15]:
```from pymks.tools import draw_goodness_of_fit
fit_data = np.array([y_train,
homogenize_model.predict(X_train, periodic_axes=[0, 1])])
pred_data = np.array([y_test, y_pred])
draw_goodness_of_fit(fit_data, pred_data, ['Training Data', 'Testing Data'])

```
```

Looks good.

The `MKSHomogenizationModel`

can be used to predict effective properties and processing-structure evolutions.

In this section of the intro, we are going to predict the local strain field in a microstructure using `MKSLocalizationModel`

, but we could have predicted another local property.

First we need some data, so let's make some.

```
In [16]:
```from pymks.datasets import make_elastic_FE_strain_delta
X_delta, y_delta = make_elastic_FE_strain_delta()

`X_delta`

is our microstructures and `y_delta`

is our local strain fields. We need to discretize the microstructure again so we will also use the same basis function.

```
In [17]:
```from pymks import MKSLocalizationModel
prim_basis = PrimitiveBasis(n_states=2)
localize_model = MKSLocalizationModel(basis=prim_basis)

Let's use the data to fit our `MKSLocalizationModel`

.

```
In [18]:
```localize_model.fit(X_delta, y_delta)

```
In [19]:
```from pymks.datasets import make_elastic_FE_strain_random
X_test, y_test = make_elastic_FE_strain_random()

Let's look at the microstructure and its local strain field.

```
In [20]:
```from pymks.tools import draw_microstructure_strain
draw_microstructure_strain(X_test[0], y_test[0])

```
```

`MKSLocalizationModel`

and compare the predicted and computed local strain field.

```
In [21]:
```from pymks.tools import draw_strains_compare
y_pred = localize_model.predict(X_test)
draw_strains_compare(y_test[0], y_pred[0])

```
```

Not bad.

The `MKSLocalizationModel`

can be used to predict local properties and local processing-structure evolutions.

```
In [ ]:
```