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Rowan Cockett
58 lines
2.1 KiB
ReStructuredText
58 lines
2.1 KiB
ReStructuredText
.. _examples_Mesh_Operators_CahnHilliard:
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.. --------------------------------- ..
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.. ..
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.. THIS FILE IS AUTO GENEREATED ..
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.. ..
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.. SimPEG/Examples/__init__.py ..
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.. ..
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.. --------------------------------- ..
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Mesh: Operators: Cahn Hilliard
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==============================
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This example is based on the example in the FiPy_ library.
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Please see their documentation for more information about the Cahn-Hilliard equation.
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The "Cahn-Hilliard" equation separates a field \\( \\phi \\) into 0 and 1 with smooth transitions.
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.. math::
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\frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \left( \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi \right)
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Where \\( f \\) is the energy function \\( f = ( a^2 / 2 )\\phi^2(1 - \\phi)^2 \\)
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which drives \\( \\phi \\) towards either 0 or 1, this competes with the term
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\\(\\epsilon^2 \\nabla^2 \\phi \\) which is a diffusion term that creates smooth changes in \\( \\phi \\).
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The equation can be factored:
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.. math::
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\frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \psi \\
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\psi = \frac{\partial^2 f}{\partial \phi^2} (\phi - \phi^{\text{old}}) + \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi
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Here we will need the derivatives of \\( f \\):
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.. math::
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\frac{\partial f}{\partial \phi} = (a^2/2)2\phi(1-\phi)(1-2\phi)
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\frac{\partial^2 f}{\partial \phi^2} = (a^2/2)2[1-6\phi(1-\phi)]
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The implementation below uses backwards Euler in time with an exponentially increasing time step.
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The initial \\( \\phi \\) is a normally distributed field with a standard deviation of 0.1 and mean of 0.5.
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The grid is 60x60 and takes a few seconds to solve ~130 times. The results are seen below, and you can see the
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field separating as the time increases.
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.. _FiPy: http://www.ctcms.nist.gov/fipy/examples/cahnHilliard/generated/examples.cahnHilliard.mesh2DCoupled.html
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.. plot::
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from SimPEG import Examples
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Examples.Mesh_Operators_CahnHilliard.run()
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.. literalinclude:: ../../SimPEG/Examples/Mesh_Operators_CahnHilliard.py
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:language: python
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:linenos:
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