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 = Mesh.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = Utils.speye(M.nC)
# Operators
A = D * M.faceDiv * M.cellGrad
L = epsilon**2 * M.faceDiv * M.cellGrad
duration = 75
elapsed = 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 = 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 = Solver(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.plotImage(PHIS[ii][1], ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: {0:4.1f}'.format(PHIS[ii][0]))
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 = Mesh.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = Utils.speye(M.nC)
# Operators
A = D * M.faceDiv * M.cellGrad
L = epsilon**2 * M.faceDiv * M.cellGrad
duration = 75
elapsed = 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 = 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 = Solver(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.plotImage(PHIS[ii][1], ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: {0:4.1f}'.format(PHIS[ii][0]))
def run(plotIt=True, n=60):
"""
Mesh: 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 \\\\( \\\\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 \\\\( f \\\\) is the energy function \\\\( f = ( a^2 / 2 )\\\\phi^2(1 - \\\\phi)^2 \\\\)
which drives \\\\( \\\\phi \\\\) towards either 0 or 1, this competes with the term
\\\\(\\\\epsilon^2 \\\\nabla^2 \\\\phi \\\\) which is a diffusion term that creates smooth changes in \\\\( \\\\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 \\\\( 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 \\\\( \\\\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: http://www.ctcms.nist.gov/fipy/examples/cahnHilliard/generated/examples.cahnHilliard.mesh2DCoupled.html
"""
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 = Mesh.TensorMesh([h, h])
# Constants
D = a = epsilon = 1.
I = Utils.speye(M.nC)
# Operators
A = D * M.faceDiv * M.cellGrad
L = epsilon**2 * M.faceDiv * M.cellGrad
duration = 75
elapsed = 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 = 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 = Solver(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.plotImage(PHIS[ii][1], ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: {0:4.1f}'.format(PHIS[ii][0]))