# Effective Stiffness¶

## Introduction¶

This example uses the MKSHomogenizationModel to create a homogenization linkage for the effective stiffness. This example starts with a brief background of the homogenization theory on the components of the effective elastic stiffness tensor for a composite material. Then the example generates random microstructures and their average stress values that will be used to show how to calibrate and use our model. We will also show how to use tools from sklearn to optimize fit parameters for the MKSHomogenizationModel. Lastly, the data is used to evaluate the MKSHomogenizationModel for effective stiffness values for a new set of microstructures.

## Linear Elasticity and Effective Elastic Modulus¶

For this example we are looking to create a homogenization linkage that predicts the effective isotropic stiffness components for two-phase microstructures. The specific stiffness component we are looking to predict in this example is $$C_{xxxx}$$ which is easily accessed by applying an uniaxial macroscal strain tensor (the only non-zero component is $$\varepsilon_{xx}$$.

$u(L, y) = u(0, y) + L\bar{\varepsilon}_{xx}$
$u(0, L) = u(0, 0) = 0$
$u(x, 0) = u(x, L)$

More details about these boundary conditions can be found in . Using these boundary conditions, $$C_{xxxx}$$ can be estimated calculating the ratio of the averaged stress over the applied averaged strain.

$C_{xxxx}^* \cong \bar{\sigma}_{xx} / \bar{\varepsilon}_{xx}$

In this example, $$C_{xxxx}$$ for 6 different types of microstructures will be estimated using the MKSHomogenizationModel from pymks, and provides a method to compute $$\bar{\sigma}_{xx}$$ for a new microstructure with an applied strain of $$\bar{\varepsilon}_{xx}$$.

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt


## Data Generation¶

A set of periodic microstructures and their volume averaged elastic stress values $$\bar{\sigma}_{xx}$$ can be generated by importing the make_elastic_stress_random function from pymks.datasets. This function has several arguments. n_samples is the number of samples that will be generated, size specifies the dimensions of the microstructures, grain_size controls the effective microstructure feature size, elastic_modulus and poissons_ratio are used to indicate the material property for each of the phases, macro_strain is the value of the applied uniaxixial strain, and the seed can be used to change the the random number generator seed.

Let’s go ahead and create 6 different types of microstructures each with 200 samples with dimensions 21 x 21. Each of the 6 samples will have a different microstructure feature size. The function will return and the microstructures and their associated volume averaged stress values.

from pymks.datasets import make_elastic_stress_random
sample_size = 200
grain_size = [(15, 2), (2, 15), (7, 7), (8, 3), (3, 9), (2, 2)]
n_samples = [sample_size] * 6
elastic_modulus = (410, 200)
poissons_ratio = (0.28, 0.3)
macro_strain = 0.001
size = (21, 21)

X, y = make_elastic_stress_random(n_samples=n_samples, size=size, grain_size=grain_size,
elastic_modulus=elastic_modulus, poissons_ratio=poissons_ratio,
macro_strain=macro_strain, seed=0)


The array X contains the microstructure information and has the dimensions of (n_samples, Nx, Ny). The array y contains the average stress value for each of the microstructures and has dimensions of (n_samples,).

print(X.shape)
print(y.shape)

(1200, 21, 21)
(1200,)


Lets take a look at the 6 types the microstructures to get an idea of what they look like. We can do this by importing draw_microstructures.

from pymks.tools import draw_microstructures
X_examples = X[::sample_size]
draw_microstructures((X_examples[:3])) draw_microstructures((X_examples[3:])) In this dataset 4 of the 6 microstructure types have grains that are elongated in either the x or y directions. The remaining 2 types of samples have equiaxed grains with different average sizes.

Let’s look at the stress values for each of the microstructures shown above.

print('Stress Values'), (y[::200])

Stress Values [ 0.30279774  0.27063703  0.30712908  0.29559632  0.28195039  0.28474614]


Now that we have a dataset to work with, we can look at how to use the MKSHomogenizationModelto predict stress values for new microstructures.

## MKSHomogenizationModel Work Flow¶

The default instance of the MKSHomogenizationModel takes in a dataset and - calculates the 2-point statistics - performs dimensionality reduction using Singular Valued Decomposition (SVD) - and fits a polynomial regression model model to the low-dimensional representation.

This work flow has been shown to accurately predict effective properties in several examples , and requires that we specify the number of components used in dimensionality reduction and the order of the polynomial we will be using for the polynomial regression. In this example we will show how we can use tools from sklearn to try and optimize our selection for these two parameters.

## Modeling with MKSHomogenizationModel¶

In order to make an instance of the MKSHomogenizationModel, we need to pass an instance of a basis (used to compute the 2-point statistics). For this particular example, there are only 2 discrete phases, so we will use the PrimitiveBasis from pymks. We only have two phases denoted by 0 and 1, therefore we have two local states and our domain is 0 to 1.

Let’s make an instance of the MKSHomgenizationModel.

from pymks import MKSHomogenizationModel
from pymks import PrimitiveBasis

prim_basis = PrimitiveBasis(n_states=2, domain=[0, 1])
model = MKSHomogenizationModel(basis=prim_basis,
correlations=[(0, 0), (1, 1), (0, 1)])


Let’s take a look at the default values for the number of components and the order of the polynomial.

print('Default Number of Components'), (model.n_components)
print('Default Polynomail Order'), (model.degree)

Default Number of Components 2
Default Polynomail Order 1


These default parameters may not be the best model for a given problem, we will now show one method that can be used to optimize them.

## Optimizing the Number of Components and Polynomial Order¶

To start with, we can look at how the variance changes as a function of the number of components. In general for SVD as well as PCA, the amount of variance captured in each component decreases as the component number increases. This means that as the number of components used in the dimensionality reduction increases, the percentage of the variance will asymptotically approach 100%. Let’s see if this is true for our dataset.

In order to do this we will change the number of components to 40 and then fit the data we have using the fit function. This function performs the dimensionality reduction and also fits the regression model. Because our microstructures are periodic, we need to use the periodic_axes argument when we fit the data.

model.n_components = 40
model.fit(X, y, periodic_axes=[0, 1])


Now look at how the cumlative variance changes as a function of the number of components using draw_component_variance from pymks.tools.

from pymks.tools import draw_component_variance

draw_component_variance(model.dimension_reducer.explained_variance_ratio_) Roughly 90 percent of the variance is captured with the first 5 components. This means our model may only need a few components to predict the average stress.

Next we need to optimize the number of components and the polynomial order. To do this we are going to split the data into testing and training sets. This can be done using the train_test_spilt function from sklearn.

from sklearn.cross_validation import train_test_split

flat_shape = (X.shape,) + (np.prod(X.shape[1:]),)

X_train, X_test, y_train, y_test = train_test_split(X.reshape(flat_shape), y,
test_size=0.2, random_state=3)
print(X_train.shape)
print(X_test.shape)

(960, 441)
(240, 441)


We will use cross validation with the testing data to fit a number of models, each with a different number of components and a different polynomial order. Then we will use the testing data to verify the best model. This can be done using GridSeachCV from sklearn.

We will pass a dictionary params_to_tune with the range of polynomial order degree and components n_components we want to try. A dictionary fit_params can be used to pass the periodic_axes variable to calculate periodic 2-point statistics. The argument cv can be used to specify the number of folds used in cross validation and n_jobs can be used to specify the number of jobs that are ran in parallel.

Let’s vary n_components from 1 to 7 and degree from 1 to 3.

from sklearn.grid_search import GridSearchCV

params_to_tune = {'degree': np.arange(1, 4), 'n_components': np.arange(1, 8)}
fit_params = {'size': X.shape, 'periodic_axes': [0, 1]}
gs = GridSearchCV(model, params_to_tune, cv=12, n_jobs=6, fit_params=fit_params).fit(X_train, y_train)


The default score method for the MKSHomogenizationModel is the R-squared value. Let’s look at the how the mean R-squared values and their standard deviations change as we varied the number of n_components and degree using draw_gridscores_matrix from pymks.tools.

from pymks.tools import draw_gridscores_matrix

draw_gridscores_matrix(gs, ['n_components', 'degree'], score_label='R-Squared',
param_labels=['Number of Components', 'Order of Polynomial']) It looks like we get a poor fit when only the first and second component are used, and when we increase the polynomial order and the components together. The models have a high standard deviation and poor R-squared values for both of these cases.

There seems to be several potential models that use 3 to 6 components. It’s difficult to see which model is the best. Let’s use our testing data X_test to see which model performs the best.

print('Order of Polynomial'), (gs.best_estimator_.degree)
print('Number of Components'), (gs.best_estimator_.n_components)
print('R-squared Value'), (gs.score(X_test, y_test))

Order of Polynomial 3
Number of Components 3
R-squared Value 0.982073916104


For the parameter range that we searched, we have found that a model with 3rd order polynomial and 3 components had the best R-squared value. It’s difficult to see the differences in the score values and the standard deviation when we have 3 or more components. Let’s take a closer look at those values using draw_grid_scores.

from pymks.tools import draw_gridscores

gs_deg_1 = [x for x in gs.grid_scores_ \
if x.parameters['degree'] == 1][2:-1]
gs_deg_2 = [x for x in gs.grid_scores_ \
if x.parameters['degree'] == 2][2:-1]
gs_deg_3 = [x for x in gs.grid_scores_ \
if x.parameters['degree'] == 3][2:-1]

draw_gridscores([gs_deg_1,  gs_deg_2, gs_deg_3], 'n_components',
data_labels=['1st Order', '2nd Order', '3rd Order'],
colors=['#f46d43', '#1a9641', '#762a83'],
param_label='Number of Components', score_label='R-Squared') As we said, a model with a 3rd order polynomial and 3 components will give us the best result, but there are several other models that will likely provide comparable results. Let’s make the best model from our grid scores.

model = gs.best_estimator_


## Prediction using MKSHomogenizationModel¶

Now that we have selected values for n_components and degree, lets fit the model with the data. Again because our microstructures are periodic, we need to use the periodic_axes argument.

model.fit(X, y, periodic_axes=[0, 1])


Let’s generate some more data that can be used to try and validate our model’s prediction accuracy. We are going to generate 20 samples of all six different types of microstructures using the same make_elastic_stress_random function.

test_sample_size = 20
n_samples = [test_sample_size] * 6
X_new, y_new = make_elastic_stress_random(n_samples=n_samples, size=size, grain_size=grain_size,
elastic_modulus=elastic_modulus, poissons_ratio=poissons_ratio,
macro_strain=macro_strain, seed=1)


Now let’s predict the stress values for the new microstructures.

y_predict = model.predict(X_new, periodic_axes=[0, 1])


We can look to see if the low-dimensional representation of the new data is similar to the low-dimensional representation of the data we used to fit the model using draw_components from pymks.tools.

from pymks.tools import draw_components

draw_components([model.reduced_fit_data[:, :2],
model.reduced_predict_data[:, :2]],
['Training Data', 'Testing Data']) The predicted data seems to be reasonably similar to the data we used to fit the model with. Now let’s look at the score value for the predicted data.

from sklearn.metrics import r2_score
print('R-squared'), (model.score(X_new, y_new, periodic_axes=[0, 1]))

R-squared 0.99834842402


Looks pretty good. Let’s print out one actual and predicted stress value for each of the 6 microstructure types to see how they compare.

print('Actual Stress   '), (y_new[::20])
print('Predicted Stress'), (y_predict[::20])

Actual Stress    [ 0.28985647  0.2831434   0.25138814  0.29399186  0.26338502  0.27548337]
Predicted Stress [ 0.29038894  0.28375754  0.25230674  0.29388488  0.26327469  0.27586485]


Lastly, we can also evaluate our prediction by looking at a goodness-of-fit plot. We can do this by importing draw_goodness_of_fit from pymks.tools.

from pymks.tools import draw_goodness_of_fit

fit_data = np.array([y, model.predict(X, periodic_axes=[0, 1])])
pred_data = np.array([y_new, y_predict])
draw_goodness_of_fit(fit_data, pred_data, ['Training Data', 'Testing Data']) We can see that the MKSHomogenizationModel has created a homogenization linkage for the effective stiffness for the 6 different microstructures and has predicted the average stress values for our new microstructures reasonably well.