examples.cahnHilliard.mesh2DCoupled

Solve the Cahn-Hilliard problem in two dimensions.

Warning

This formulation has serious performance problems and is not automatically tested. Specifically, for non-trivial mesh sizes, PySparse requires enormous amounts of memory, Trilinos cannot solve the coupled form, and PETSc cannot solve the vector form.

The spinodal decomposition phenomenon is a spontaneous separation of an initially homogeneous mixture into two distinct regions of different properties (spin-up/spin-down, component A/component B). It is a “barrierless” phase separation process, such that under the right thermodynamic conditions, any fluctuation, no matter how small, will tend to grow. This is in contrast to nucleation, where a fluctuation must exceed some critical magnitude before it will survive and grow. Spinodal decomposition can be described by the “Cahn-Hilliard” equation (also known as “conserved Ginsberg-Landau” or “model B” of Hohenberg & Halperin)

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

where \phi is a conserved order parameter, possibly representing alloy composition or spin. The double-well free energy function f = (a^2/2) \phi^2 (1 - \phi)^2 penalizes states with intermediate values of \phi between 0 and 1. The gradient energy term \epsilon^2 \nabla^2\phi, on the other hand, penalizes sharp changes of \phi. These two competing effects result in the segregation of \phi into domains of 0 and 1, separated by abrupt, but smooth, transitions. The parameters a and \epsilon determine the relative weighting of the two effects and D is a rate constant.

We can simulate this process in FiPy with a simple script:

>>> from fipy import CellVariable, Grid2D, GaussianNoiseVariable, DiffusionTerm, TransientTerm, ImplicitSourceTerm, Viewer
>>> from fipy.tools import numerix

(Note that all of the functionality of NumPy is imported along with FiPy, although much is augmented for FiPy's needs.)

>>> if __name__ == "__main__":
...     nx = ny = 20
... else:
...     nx = ny = 10
>>> mesh = Grid2D(nx=nx, ny=ny, dx=0.25, dy=0.25)
>>> phi = CellVariable(name=r"$\phi$", mesh=mesh)
>>> psi = CellVariable(name=r"$\psi$", mesh=mesh)

We start the problem with random fluctuations about \phi = 1/2

>>> noise = GaussianNoiseVariable(mesh=mesh,
...                               mean=0.5,
...                               variance=0.01).value
>>> phi[:] = noise

FiPy doesn’t plot or output anything unless you tell it to:

>>> if __name__ == "__main__":
...     viewer = Viewer(vars=(phi,), datamin=0., datamax=1.)

We factor the Cahn-Hilliard equation into two 2nd-order PDEs and place them in canonical form for FiPy to solve them as a coupled set of equations.

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

The source term in \psi, \frac{\partial f}{\partial \phi}, is expressed in linearized form after Taylor expansion at \phi=\phi_{\text{old}}, for the same reasons discussed in examples.phase.simple. We need to perform the partial derivatives

\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 \left[1 - 6 \phi(1 - \phi)\right]

manually.

>>> D = a = epsilon = 1.
>>> dfdphi = a**2 * phi * (1 - phi) * (1 - 2 * phi)
>>> dfdphi_ = a**2 * (1 - phi) * (1 - 2 * phi)
>>> d2fdphi2 = a**2 * (1 - 6 * phi * (1 - phi))
>>> eq1 = (TransientTerm(var=phi) == DiffusionTerm(coeff=D, var=psi))
>>> eq2 = (ImplicitSourceTerm(coeff=1., var=psi)
...        == ImplicitSourceTerm(coeff=d2fdphi2, var=phi) - d2fdphi2 * phi + dfdphi
...        - DiffusionTerm(coeff=epsilon**2, var=phi))
>>> eq3 = (ImplicitSourceTerm(coeff=1., var=psi)
...        == ImplicitSourceTerm(coeff=dfdphi_, var=phi)
...        - DiffusionTerm(coeff=epsilon**2, var=phi))
>>> eq = eq1 & eq2

Because the evolution of a spinodal microstructure slows with time, we use exponentially increasing time steps to keep the simulation “interesting”. The FiPy user always has direct control over the evolution of their problem.

>>> dexp = -5
>>> elapsed = 0.
>>> if __name__ == "__main__":
...     duration = 100.
... else:
...     duration = .5e-1
>>> while elapsed < duration:
...     dt = min(100, numerix.exp(dexp))
...     elapsed += dt
...     dexp += 0.01
...     eq.solve(dt=dt)
...     if __name__ == "__main__":
...         viewer.plot()
>>> from fipy import input
>>> if __name__ == '__main__':
...     input("Coupled equations. Press <return> to proceed...")
../../../_images/mesh2DCoupled.png

These equations can also be solved in FiPy using a vector equation. The variables \phi and \psi are now stored in a single variable

>>> var = CellVariable(mesh=mesh, elementshape=(2,))
>>> var[0] = noise
>>> if __name__ == "__main__":
...     viewer = Viewer(name=r"$\phi$", vars=var[0,], datamin=0., datamax=1.)
>>> D = a = epsilon = 1.
>>> v0 = var[0]
>>> dfdphi = a**2 * v0 * (1 - v0) * (1 - 2 * v0)
>>> dfdphi_ = a**2 * (1 - v0) * (1 - 2 * v0)
>>> d2fdphi2 = a**2 * (1 - 6 * v0 * (1 - v0))

The source terms have to be shaped correctly for a vector. The implicit source coefficient has to have a shape of (2, 2) while the explicit source has a shape (2,)

>>> source = (- d2fdphi2 * v0 + dfdphi) * (0, 1)
>>> impCoeff = d2fdphi2 * ((0, 0),
...                        (1., 0)) + ((0, 0),
...                                    (0, -1.))

This is the same equation as the previous definition of eq, but now in a vector format.

>>> eq = (TransientTerm(((1., 0.),
...                     (0., 0.))) == DiffusionTerm([((0.,          D),
...                                                   (-epsilon**2, 0.))])
...                                   + ImplicitSourceTerm(impCoeff) + source)
>>> dexp = -5
>>> elapsed = 0.
>>> while elapsed < duration:
...     dt = min(100, numerix.exp(dexp))
...     elapsed += dt
...     dexp += 0.01
...     eq.solve(var=var, dt=dt)
...     if __name__ == "__main__":
...         viewer.plot()
>>> print(numerix.allclose(var, (phi, psi)))
True
Last updated on Jan 14, 2021. Created using Sphinx 3.4.3.