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New examples for Mesh

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Rowan Cockett
2015-11-26 13:56:04 -07:00
parent be3667b4ab
commit b2aab4163b
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SimPEG/Examples/Mesh_Operators_CahnHilliard.py
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from SimPEG import *
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:
import matplotlib.pyplot as plt
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)
out = M.plotImage(PHIS[ii][1],ax=ax)
ax.axis('off')
ax.set_title('Elapsed Time: %4.1f'%PHIS[ii][0])
plt.show()
if __name__ == '__main__':
run()
SimPEG/Examples/Mesh_QuadTree_FaceDiv.py
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from SimPEG import *
def run(plotIt=True, n=60):
"""
Mesh: QuadTree: FaceDiv
=======================
"""
M = Mesh.TreeMesh([[(1,16)],[(1,16)]], levels=4)
M._refineCell([0,0,0])
M._refineCell([0,0,1])
M._refineCell([4,4,2])
M.__dirty__ = True
M.number()
if plotIt:
import matplotlib.pyplot as plt
fig, axes = plt.subplots(2,1,figsize=(10,10))
M.plotGrid(cells=True, nodes=False, ax=axes[0])
axes[0].axis('off')
axes[0].set_title('Simple QuadTree Mesh')
axes[0].set_xlim([-1,17])
axes[0].set_ylim([-1,17])
for ii, loc in zip(range(M.nC),M.gridCC):
axes[0].text(loc[0]+0.2,loc[1],'%d'%ii, color='r')
axes[0].plot(M.gridFx[:,0],M.gridFx[:,1], 'g>')
for ii, loc in zip(range(M.nFx),M.gridFx):
axes[0].text(loc[0]+0.2,loc[1],'%d'%ii, color='g')
axes[0].plot(M.gridFy[:,0],M.gridFy[:,1], 'm^')
for ii, loc in zip(range(M.nFy),M.gridFy):
axes[0].text(loc[0]+0.2,loc[1]+0.2,'%d'%(ii+M.nFx), color='m')
axes[1].spy(M.faceDiv)
axes[1].set_title('Face Divergence')
axes[1].set_ylabel('Cell Number')
axes[1].set_xlabel('Face Number')
plt.show()
if __name__ == '__main__':
run()
docs/examples/Mesh_Operators_CahnHilliard.rst
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.. _examples_Mesh_Operators_CahnHilliard:
.. --------------------------------- ..
.. ..
.. THIS FILE IS AUTO GENEREATED ..
.. ..
.. SimPEG/Examples/__init__.py ..
.. ..
.. --------------------------------- ..
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
.. plot::
from SimPEG import Examples
Examples.Mesh_Operators_CahnHilliard.run()
.. literalinclude:: ../../SimPEG/Examples/Mesh_Operators_CahnHilliard.py
:language: python
:linenos:
docs/examples/Mesh_QuadTree_FaceDiv.rst
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.. _examples_Mesh_QuadTree_FaceDiv:
.. --------------------------------- ..
.. ..
.. THIS FILE IS AUTO GENEREATED ..
.. ..
.. SimPEG/Examples/__init__.py ..
.. ..
.. --------------------------------- ..
Mesh: QuadTree: FaceDiv
=======================
.. plot::
from SimPEG import Examples
Examples.Mesh_QuadTree_FaceDiv.run()
.. literalinclude:: ../../SimPEG/Examples/Mesh_QuadTree_FaceDiv.py
:language: python
:linenos: