# 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) where is a conserved order parameter, possibly representing alloy composition or spin. The double-well free energy function penalizes states with intermediate values of between 0 and 1. The gradient energy term , on the other hand, penalizes sharp changes of . These two competing effects result in the segregation of into domains of 0 and 1, separated by abrupt, but smooth, transitions. The parameters and determine the relative weighting of the two effects and 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 >>> 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. The source term in , , is expressed in linearized form after Taylor expansion at , for the same reasons discussed in examples.phase.simple. We need to perform the partial derivatives 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...") These equations can also be solved in FiPy using a vector equation. The variables and are now stored in a single variable

>>> var = CellVariable(mesh=mesh, elementshape=(2,))
>>> var = noise

>>> if __name__ == "__main__":
...     viewer = Viewer(name=r"$\phi$", vars=var[0,], datamin=0., datamax=1.)

>>> D = a = epsilon = 1.
>>> v0 = var
>>> 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.