r"""
Operators: Cahn Hilliard
========================

This example is based on the example in the FiPy_ library.
Please see their documentation for more information about the
Cahn-Hilliard equation.

The "Cahn-Hilliard" equation separates a field :math:`\phi`
into 0 and 1 with smooth transitions.

.. math::

    \frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \left( \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi \right)

Where :math:`f` is the energy function :math:`f = ( a^2 / 2 )\phi^2(1 - \phi)^2`
which drives :math:`\phi` towards either 0 or 1, this competes with the term
:math:`\epsilon^2 \nabla^2 \phi` which is a diffusion term that creates smooth changes in :math:`\phi`.
The equation can be factored:

.. math::

    \frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \psi \\
    \psi = \frac{\partial^2 f}{\partial \phi^2} (\phi - \phi^{\text{old}}) + \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi

Here we will need the derivatives of :math:`f`:

.. math::

    \frac{\partial f}{\partial \phi} = (a^2/2)2\phi(1-\phi)(1-2\phi)
    \frac{\partial^2 f}{\partial \phi^2} = (a^2/2)2[1-6\phi(1-\phi)]

The implementation below uses backwards Euler in time with an
exponentially increasing time step. The initial :math:`\phi`
is a normally distributed field with a standard deviation of 0.1 and
mean of 0.5. The grid is 60x60 and takes a few seconds to solve ~130
times. The results are seen below, and you can see the field separating
as the time increases.

.. _FiPy: https://github.com/usnistgov/fipy

.. http://www.ctcms.nist.gov/fipy/examples/cahnHilliard/generated/examples.cahnHilliard.mesh2DCoupled.html

"""

import discretize
import numpy as np
import matplotlib.pyplot as plt
from scipy.sparse.linalg import spsolve


def run(plotIt=True, n=60):
    np.random.seed(5)

    # Here we are going to rearrange the equations:

    # (phi_ - phi)/dt = A*(d2fdphi2*(phi_ - phi) + dfdphi - L*phi_)
    # (phi_ - phi)/dt = A*(d2fdphi2*phi_ - d2fdphi2*phi + dfdphi - L*phi_)
    # (phi_ - phi)/dt = A*d2fdphi2*phi_ + A*( - d2fdphi2*phi + dfdphi - L*phi_)
    # phi_ - phi = dt*A*d2fdphi2*phi_ + dt*A*(- d2fdphi2*phi + dfdphi - L*phi_)
    # phi_ - dt*A*d2fdphi2 * phi_ =  dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
    # (I - dt*A*d2fdphi2) * phi_ =  dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
    # (I - dt*A*d2fdphi2) * phi_ =  dt*A*dfdphi - dt*A*d2fdphi2*phi - dt*A*L*phi_ + phi
    # (dt*A*d2fdphi2 - I) * phi_ =  dt*A*d2fdphi2*phi + dt*A*L*phi_ - phi - dt*A*dfdphi
    # (dt*A*d2fdphi2 - I - dt*A*L) * phi_ =  (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi

    h = [(0.25, n)]
    M = discretize.TensorMesh([h, h])

    # Constants
    D = a = epsilon = 1.0
    I = discretize.utils.speye(M.nC)

    # Operators
    A = D * M.face_divergence * M.cell_gradient
    L = epsilon**2 * M.face_divergence * M.cell_gradient

    duration = 75
    elapsed = 0.0
    dexp = -5
    phi = np.random.normal(loc=0.5, scale=0.01, size=M.nC)
    ii, jj = 0, 0
    PHIS = []
    capture = np.logspace(-1, np.log10(duration), 8)
    while elapsed < duration:
        dt = min(100, np.exp(dexp))
        elapsed += dt
        dexp += 0.05

        dfdphi = a**2 * 2 * phi * (1 - phi) * (1 - 2 * phi)
        d2fdphi2 = discretize.utils.sdiag(a**2 * 2 * (1 - 6 * phi * (1 - phi)))

        MAT = dt * A * d2fdphi2 - I - dt * A * L
        rhs = (dt * A * d2fdphi2 - I) * phi - dt * A * dfdphi
        phi = spsolve(MAT, rhs)

        if elapsed > capture[jj]:
            PHIS += [(elapsed, phi.copy())]
            jj += 1
        if ii % 10 == 0:
            print(ii, elapsed)
        ii += 1

    if plotIt:
        fig, axes = plt.subplots(2, 4, figsize=(14, 6))
        axes = np.array(axes).flatten().tolist()
        for ii, ax in zip(np.linspace(0, len(PHIS) - 1, len(axes)), axes):
            ii = int(ii)
            M.plot_image(PHIS[ii][1], ax=ax)
            ax.axis("off")
            ax.set_title("Elapsed Time: {0:4.1f}".format(PHIS[ii][0]))


if __name__ == "__main__":
    run()
    plt.show()