%matplotlib inline
%load_ext autoreload
%autoreload 2

import numpy as np
import matplotlib.pyplot as plt

from pymks.datasets import make_delta_microstructures
n = 21
n_phases = 3



"""
Explanation: Linear Elasticity in 2D for 3 Phases
Introduction
This example provides a demonstration of using PyMKS to compute the linear strain field for a three phase composite material. It demonstrates how to generate data for delta microstructures and then use this data to calibrate the first order MKS influence coefficients. The calibrated influence coefficients are used to predict the strain response for a random microstructure and the results are compared with those from finite element. Finally, the influence coefficients are scaled up and the MKS results are again compared with the finite element data for a large problem.
PyMKS uses the finite element tool SfePy to generate both the strain fields to fit the MKS model and the verification data to evaluate the MKS model's accuracy.
Elastostatics Equations and Boundary Conditions
The governing equations for elasticostaics and the boundary conditions used in this example are the same as those provided in the Linear Elastic in 2D example. 
Note that an inappropriate boundary condition is used in this example because current version of SfePy is unable to implement a periodic plus displacement boundary condition. This leads to some issues near the edges of the domain and introduces errors into the resizing of the coefficients. We are working to fix this issue, but note that the problem is not with the MKS regression itself, but with the calibration data used. The finite element package ABAQUS includes the displaced periodic boundary condition and can be used to calibrate the MKS regression correctly.
Modeling with MKS
Calibration Data and Delta Microstructures
The first order MKS influence coefficients are all that is needed to compute a strain field of a random microstructure as long as the ratio between the elastic moduli (also known as the contrast) is less than 1.5. If this condition is met we can expect a mean absolute error of 2%  or less when comparing the MKS results with those computed using finite element methods [1]. 
Because we are using distinct phases and the contrast is low enough to only need the first order coefficients, delta microstructures and their strain fields are all that we need to calibrate the first order influence coefficients [2]. 
Here we use the make_delta_microstructure function from pymks.datasets to create the delta microstructures needed to calibrate the first order influence coefficients for a two phase microstructure. The make_delta_microstructure function uses SfePy to generate the data
End of explanation
"""


from pymks.tools import draw_microstructures



"""
Explanation: Let's take a look at a few of the delta microstructures by importing draw_microstructures from pymks.tools.
End of explanation
"""


from pymks.datasets import make_elastic_FE_strain_delta

elastic_modulus = (80, 100, 120)
poissons_ratio = (0.3, 0.3, 0.3)
macro_strain = 0.02
size = (n, n)

= make_elastic_FE_strain_delta(elastic_modulus=elastic_modulus,
                                                      poissons_ratio=poissons_ratio,
                                                      size=size, macro_strain=macro_strain)

"""
Explanation: Using delta microstructures for the calibration of the first order influence coefficients is essentially the same as using a unit impulse response to find the kernel of a system in signal processing. Any given delta microstructure is composed of only two phases with the center cell having an alternative phase from the remainder of the domain. The number of delta microstructures that are needed to calibrated the first order coefficients is $N(N-1)$ where $N$ is the number of phases, therefore in this example we need 6 delta microstructures.
Generating Calibration Data
The make_elasticFEstrain_delta function from pymks.datasets provides an easy interface to generate delta microstructures and their strain fields, which can then be used for calibration of the influence coefficients. The function calls the ElasticFESimulation class to compute the strain fields.
In this example, lets look at a three phase microstructure with elastic moduli values of 80, 100 and 120 and Poisson's ratio values all equal to 0.3. Let's also set the macroscopic imposed strain equal to 0.02. All of these parameters used in the simulation must be passed into the make_elasticFEstrain_delta function. The number of Poisson's ratio values and elastic moduli values indicates the number of phases. Note that make_elasticFEstrain_delta does not take a number of samples argument as the number of samples to calibrate the MKS is fixed by the number of phases.
End of explanation
"""


from pymks.tools import draw_microstructure_strain


"""
Explanation: Let's take a look at one of the delta microstructures and the $\varepsilon_{xx}$ strain field using the draw_microstructure_strain from pymks.tools.
End of explanation
"""


from pymks import MKSLocalizationModel
from pymks import PrimitiveBasis


"""
Explanation: Because slice(None) (the default slice operator in Python, equivalent to array[:]) was passed in to the make_elasticFEstrain_delta function as the argument for strain_index, the function returns all the strain fields. Let's also take a look at the $\varepsilon_{yy}$ and $\varepsilon_{xy}$ strain fields.
Calibrating First Order Influence Coefficients
Now that we have the delta microstructures and their strain fields, we can calibrate the influence coefficients by creating an instance of the MKSLocalizationModel class and PrimitiveBasis class. Because we have 3 phases and we know their values range from [0, 2], we will create an instance of PrimitiveBasis with n_states equal to 3 and domain equal to [0, 2].
Next we will create an instance of the MKSLocalizationModel with basis set equal to the instance of PrimitiveBasis we create.
End of explanation
"""


from pymks.tools import draw_coeff


"""
Explanation: Now, pass the delta microstructures and their strain fields into the fit method to calibrate the first order influence coefficients.
That's it, the influence coefficient have be calibrated. Let's take a look at them.
End of explanation
"""


from pymks.datasets import make_elastic_FE_strain_random

np.random.seed(101)

"""
Explanation: The influence coefficients for $l=0$ and $l = 1$ have a Gaussian-like shape, while the influence coefficients for $l=2$ are constant-valued. The constant-valued influence coefficients may seem superfluous, but are equally as import. They are equivalent to the constant term in multiple linear regression with categorical variables.
Predict of the Strain Field for a Random Microstructure
Let's now use our instance of the MKSLocalizationModel class with calibrated influence coefficients to compute the strain field for a random two phase microstructure and compare it with the results from a finite element simulation. 
The make_elasticFEstrain_random function from pymks.datasets is an easy way to generate a random microstructure and its strain field results from finite element analysis.
End of explanation
"""


from pymks.tools import draw_strains_compare



"""
Explanation: Note that the calibrated influence coefficients can only be used to reproduce the simulation with the same boundary conditions that they were calibrated with
Let's take a look at the microstructure and the strain field using the draw_microstructure_strain function.
Now to get the strain field from the MKSLocalizationModel just pass the same microstructure to the predict method.
Finally let's compare the results from finite element simulation and the MKS model,using draw_strains_compare from pymks.tools.
End of explanation
"""


from pymks.tools import draw_differences


"""
Explanation: Let's plot the difference between the two strain fields, using draw_differences from pymks.tools.
End of explanation
"""


m = 3 * n 
size = (m, m)
print size

= make_elastic_FE_strain_random(n_samples=1, elastic_modulus=elastic_modulus,
                                          poissons_ratio=poissons_ratio, size=size,
                                          macro_strain=macro_strain)

"""
Explanation: The MKS model is able to capture the strain field for the random microstructure after being calibrated with delta microstructures.
Resizing the Coefficeints to use on Larger Microstructures
The influence coefficients that were calibrated on a smaller microstructure can be used to predict the strain field on a larger microstructure though spectral interpolation [3], but accuracy of the MKS model drops slightly. To demonstrate how this is done, let's generate a new larger random microstructure and its strain field.
End of explanation
"""


