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https://github.com/wassname/simpeg.git
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Rowan Cockett
106 lines
4.0 KiB
Python
106 lines
4.0 KiB
Python
from SimPEG import *
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def run(plotIt=True, n=60):
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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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"""
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np.random.seed(5)
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# Here we are going to rearrange the equations:
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# (phi_ - phi)/dt = A*(d2fdphi2*(phi_ - phi) + dfdphi - L*phi_)
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# (phi_ - phi)/dt = A*(d2fdphi2*phi_ - d2fdphi2*phi + dfdphi - L*phi_)
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# (phi_ - phi)/dt = A*d2fdphi2*phi_ + A*( - d2fdphi2*phi + dfdphi - L*phi_)
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# phi_ - phi = dt*A*d2fdphi2*phi_ + dt*A*(- d2fdphi2*phi + dfdphi - L*phi_)
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# phi_ - dt*A*d2fdphi2 * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
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# (I - dt*A*d2fdphi2) * phi_ = dt*A*(- d2fdphi2*phi + dfdphi - L*phi_) + phi
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# (I - dt*A*d2fdphi2) * phi_ = dt*A*dfdphi - dt*A*d2fdphi2*phi - dt*A*L*phi_ + phi
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# (dt*A*d2fdphi2 - I) * phi_ = dt*A*d2fdphi2*phi + dt*A*L*phi_ - phi - dt*A*dfdphi
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# (dt*A*d2fdphi2 - I - dt*A*L) * phi_ = (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi
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h = [(0.25,n)]
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M = Mesh.TensorMesh([h,h])
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# Constants
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D = a = epsilon = 1.
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I = Utils.speye(M.nC)
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# Operators
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A = D * M.faceDiv * M.cellGrad
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L = epsilon**2 * M.faceDiv * M.cellGrad
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duration = 75
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elapsed = 0.
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dexp = -5
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phi = np.random.normal(loc=0.5,scale=0.01,size=M.nC)
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ii, jj = 0, 0
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PHIS = []
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capture = np.logspace(-1,np.log10(duration),8)
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while elapsed < duration:
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dt = min(100, np.exp(dexp))
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elapsed += dt
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dexp += 0.05
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dfdphi = a**2 * 2 * phi * (1 - phi) * (1 - 2 * phi)
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d2fdphi2 = Utils.sdiag(a**2 * 2 * (1 - 6 * phi * (1 - phi)))
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MAT = (dt*A*d2fdphi2 - I - dt*A*L)
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rhs = (dt*A*d2fdphi2 - I)*phi - dt*A*dfdphi
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phi = Solver(MAT)*rhs
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if elapsed > capture[jj]:
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PHIS += [(elapsed, phi.copy())]
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jj += 1
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if ii % 10 == 0: print ii, elapsed
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ii += 1
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if plotIt:
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import matplotlib.pyplot as plt
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fig, axes = plt.subplots(2,4,figsize=(14,6))
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axes = np.array(axes).flatten().tolist()
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for ii, ax in zip(np.linspace(0,len(PHIS)-1,len(axes)),axes):
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ii = int(ii)
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out = M.plotImage(PHIS[ii][1],ax=ax)
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ax.axis('off')
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ax.set_title('Elapsed Time: %4.1f'%PHIS[ii][0])
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plt.show()
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if __name__ == '__main__':
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run()
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