Cahn-Hilliard equation
======================

This demo is implemented in a single Python file,
:download:`demo_cahn-hilliard.py`, which contains both the variational
forms and the solver.

This example demonstrates the solution of a particular nonlinear
time-dependent fourth-order equation, known as the Cahn-Hilliard
equation. In particular it demonstrates the use of

* The built-in Newton solver
* Advanced use of the base class ``NonlinearProblem``
* Automatic linearisation
* A mixed finite element method
* The :math:`\theta`-method for time-dependent equations
* User-defined Expressions as Python classes
* Form compiler options
* Interpolation of functions


Equation and problem definition
-------------------------------

The Cahn-Hilliard equation is a parabolic equation and is typically
used to model phase separation in binary mixtures.  It involves
first-order time derivatives, and second- and fourth-order spatial
derivatives.  The equation reads:

.. math::
   \frac{\partial c}{\partial t} - \nabla \cdot M \left(\nabla\left(\frac{d f}{d c}
             - \lambda \nabla^{2}c\right)\right) &= 0 \quad {\rm in} \ \Omega, \\
   M\left(\nabla\left(\frac{d f}{d c} - \lambda \nabla^{2}c\right)\right) \cdot n &= 0 \quad {\rm on} \ \partial\Omega, \\
   M \lambda \nabla c \cdot n &= 0 \quad {\rm on} \ \partial\Omega.

where :math:`c` is the unknown field, the function :math:`f` is
usually non-convex in :math:`c` (a fourth-order polynomial is commonly
used), :math:`n` is the outward directed boundary normal, and
:math:`M` is a scalar parameter.


Mixed form
^^^^^^^^^^

The Cahn-Hilliard equation is a fourth-order equation, so casting it
in a weak form would result in the presence of second-order spatial
derivatives, and the problem could not be solved using a standard
Lagrange finite element basis.  A solution is to rephrase the problem
as two coupled second-order equations:

.. math::
   \frac{\partial c}{\partial t} - \nabla \cdot M \nabla\mu  &= 0 \quad {\rm in} \ \Omega, \\
   \mu -  \frac{d f}{d c} + \lambda \nabla^{2}c &= 0 \quad {\rm in} \ \Omega.

The unknown fields are now :math:`c` and :math:`\mu`. The weak
(variational) form of the problem reads: find :math:`(c, \mu) \in V
\times V` such that

.. math::
   \int_{\Omega} \frac{\partial c}{\partial t} q \, {\rm d} x + \int_{\Omega} M \nabla\mu \cdot \nabla q \, {\rm d} x
          &= 0 \quad \forall \ q \in V,  \\
   \int_{\Omega} \mu v \, {\rm d} x - \int_{\Omega} \frac{d f}{d c} v \, {\rm d} x - \int_{\Omega} \lambda \nabla c \cdot \nabla v \, {\rm d} x
          &= 0 \quad \forall \ v \in V.


Time discretisation
^^^^^^^^^^^^^^^^^^^

Before being able to solve this problem, the time derivative must be
dealt with. Apply the :math:`\theta`-method to the mixed weak form of
the equation:

.. math::

   \int_{\Omega} \frac{c_{n+1} - c_{n}}{dt} q \, {\rm d} x + \int_{\Omega} M \nabla \mu_{n+\theta} \cdot \nabla q \, {\rm d} x
          &= 0 \quad \forall \ q \in V  \\
   \int_{\Omega} \mu_{n+1} v  \, {\rm d} x - \int_{\Omega} \frac{d f_{n+1}}{d c} v  \, {\rm d} x - \int_{\Omega} \lambda \nabla c_{n+1} \cdot \nabla v \, {\rm d} x
          &= 0 \quad \forall \ v \in V

where :math:`dt = t_{n+1} - t_{n}` and :math:`\mu_{n+\theta} =
(1-\theta) \mu_{n} + \theta \mu_{n+1}`.  The task is: given
:math:`c_{n}` and :math:`\mu_{n}`, solve the above equation to find
:math:`c_{n+1}` and :math:`\mu_{n+1}`.


Demo parameters
^^^^^^^^^^^^^^^

The following domains, functions and time stepping parameters are used
in this demo:

* :math:`\Omega = (0, 1) \times (0, 1)` (unit square)
* :math:`f = 100 c^{2} (1-c)^{2}`
* :math:`\lambda = 1 \times 10^{-2}`
* :math:`M = 1`
* :math:`dt = 5 \times 10^{-6}`
* :math:`\theta = 0.5`


Implementation
--------------

This demo is implemented in the :download:`demo_cahn-hilliard.py`
file.

First, the modules :py:mod:`random` :py:mod:`matplotlib`
:py:mod:`dolfin` module are imported::

    import random
    from dolfin import *

.. index:: Expression

A class which will be used to represent the initial conditions is then
created::

    # Class representing the intial conditions
    class InitialConditions(UserExpression):
        def __init__(self, **kwargs):
            random.seed(2 + MPI.rank(MPI.comm_world))
            super().__init__(**kwargs)
        def eval(self, values, x):
            values[0] = 0.63 + 0.02*(0.5 - random.random())
            values[1] = 0.0
        def value_shape(self):
            return (2,)

It is a subclass of :py:class:`Expression
<dolfin.functions.expression.Expression>`. In the constructor
(``__init__``), the random number generator is seeded. If the program
is run in parallel, the random number generator is seeded using the
rank (process number) to ensure a different sequence of numbers on
each process.  The function ``eval`` returns values for a function of
dimension two.  For the first component of the function, a randomized
value is returned.  The method ``value_shape`` declares that the
:py:class:`Expression <dolfin.functions.expression.Expression>` is
vector valued with dimension two.

.. index::
   single: NonlinearProblem; (in Cahn-Hilliard demo)

A class which will represent the Cahn-Hilliard in an abstract from for
use in the Newton solver is now defined. It is a subclass of
:py:class:`NonlinearProblem <dolfin.cpp.NonlinearProblem>`. ::

    # Class for interfacing with the Newton solver
    class CahnHilliardEquation(NonlinearProblem):
        def __init__(self, a, L):
            NonlinearProblem.__init__(self)
            self.L = L
            self.a = a
        def F(self, b, x):
            assemble(self.L, tensor=b)
        def J(self, A, x):
            assemble(self.a, tensor=A)

The constructor (``__init__``) stores references to the bilinear
(``a``) and linear (``L``) forms. These will used to compute the
Jacobian matrix and the residual vector, respectively, for use in a
Newton solver.  The function ``F`` and ``J`` are virtual member
functions of :py:class:`NonlinearProblem
<dolfin.cpp.NonlinearProblem>`. The function ``F`` computes the
residual vector ``b``, and the function ``J`` computes the Jacobian
matrix ``A``.

Next, various model parameters are defined::

    # Model parameters
    lmbda  = 1.0e-02  # surface parameter
    dt     = 5.0e-06  # time step
    theta  = 0.5      # time stepping family, e.g. theta=1 -> backward Euler, theta=0.5 -> Crank-Nicolson

.. index::
   singe: form compiler options; (in Cahn-Hilliard demo)

It is possible to pass arguments that control aspects of the generated
code to the form compiler. The lines ::

    # Form compiler options
    parameters["form_compiler"]["optimize"]     = True
    parameters["form_compiler"]["cpp_optimize"] = True

tell the form to apply optimization strategies in the code generation
phase and the use compiler optimization flags when compiling the
generated C++ code. Using the option ``["optimize"] = True`` will
generally result in faster code (sometimes orders of magnitude faster
for certain operations, depending on the equation), but it may take
considerably longer to generate the code and the generation phase may
use considerably more memory).

A unit square mesh with 97 (= 96 + 1) vertices in each direction is
created, and on this mesh a :py:class:`FunctionSpace
<dolfin.functions.functionspace.FunctionSpace>` ``ME`` is built using
a pair of linear Lagrangian elements. ::

    # Create mesh and build function space
    mesh = UnitSquareMesh.create(96, 96, CellType.Type.quadrilateral)
    P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
    ME = FunctionSpace(mesh, P1*P1)

Trial and test functions of the space ``ME`` are now defined::

    # Define trial and test functions
    du    = TrialFunction(ME)
    q, v  = TestFunctions(ME)

.. index:: split functions

For the test functions, :py:func:`TestFunctions
<dolfin.functions.function.TestFunctions>` (note the 's' at the end)
is used to define the scalar test functions ``q`` and ``v``. The
:py:class:`TrialFunction <dolfin.functions.function.TrialFunction>`
``du`` has dimension two. Some mixed objects of the
:py:class:`Function <dolfin.functions.function.Function>` class on
``ME`` are defined to represent :math:`u = (c_{n+1}, \mu_{n+1})` and
:math:`u0 = (c_{n}, \mu_{n})`, and these are then split into
sub-functions::

    # Define functions
    u   = Function(ME)  # current solution
    u0  = Function(ME)  # solution from previous converged step

    # Split mixed functions
    dc, dmu = split(du)
    c,  mu  = split(u)
    c0, mu0 = split(u0)

The line ``c, mu = split(u)`` permits direct access to the components
of a mixed function. Note that ``c`` and ``mu`` are references for
components of ``u``, and not copies.

.. index::
   single: interpolating functions; (in Cahn-Hilliard demo)

Initial conditions are created by using the class defined at the
beginning of the demo and then interpolating the initial conditions
into a finite element space::

    # Create intial conditions and interpolate
    u_init = InitialConditions(degree=1)
    u.interpolate(u_init)
    u0.interpolate(u_init)

The first line creates an object of type ``InitialConditions``.  The
following two lines make ``u`` and ``u0`` interpolants of ``u_init``
(since ``u`` and ``u0`` are finite element functions, they may not be
able to represent a given function exactly, but the function can be
approximated by interpolating it in a finite element space).

.. index:: automatic differentiation

The chemical potential :math:`df/dc` is computed using automated
differentiation::

    # Compute the chemical potential df/dc
    c = variable(c)
    f    = 100*c**2*(1-c)**2
    dfdc = diff(f, c)

The first line declares that ``c`` is a variable that some function
can be differentiated with respect to. The next line is the function
:math:`f` defined in the problem statement, and the third line
performs the differentiation of ``f`` with respect to the variable
``c``.

It is convenient to introduce an expression for :math:`\mu_{n+\theta}`::

    # mu_(n+theta)
    mu_mid = (1.0-theta)*mu0 + theta*mu

which is then used in the definition of the variational forms::

    # Weak statement of the equations
    L0 = c*q*dx - c0*q*dx + dt*dot(grad(mu_mid), grad(q))*dx
    L1 = mu*v*dx - dfdc*v*dx - lmbda*dot(grad(c), grad(v))*dx
    L = L0 + L1

This is a statement of the time-discrete equations presented as part
of the problem statement, using UFL syntax. The linear forms for the
two equations can be summed into one form ``L``, and then the
directional derivative of ``L`` can be computed to form the bilinear
form which represents the Jacobian matrix::

    # Compute directional derivative about u in the direction of du (Jacobian)
    a = derivative(L, u, du)

.. index::
   single: Newton solver; (in Cahn-Hilliard demo)

The DOLFIN Newton solver requires a :py:class:`NonlinearProblem
<dolfin.cpp.NonlinearProblem>` object to solve a system of nonlinear
equations. Here, we are using the class ``CahnHilliardEquation``,
which was declared at the beginning of the file, and which is a
sub-class of :py:class:`NonlinearProblem
<dolfin.cpp.NonlinearProblem>`. We need to instantiate objects of both
``CahnHilliardEquation`` and :py:class:`NewtonSolver
<dolfin.cpp.NewtonSolver>`::

    # Create nonlinear problem and Newton solver
    problem = CahnHilliardEquation(a, L)
    solver = NewtonSolver()
    solver.parameters["linear_solver"] = "lu"
    solver.parameters["convergence_criterion"] = "incremental"
    solver.parameters["relative_tolerance"] = 1e-6

The string ``"lu"`` passed to the Newton solver indicated that an LU
solver should be used.  The setting of
``parameters["convergence_criterion"] = "incremental"`` specifies that
the Newton solver should compute a norm of the solution increment to
check for convergence (the other possibility is to use ``"residual"``,
or to provide a user-defined check). The tolerance for convergence is
specified by ``parameters["relative_tolerance"] = 1e-6``.

To run the solver and save the output to a VTK file for later visualization,
the solver is advanced in time from :math:`t_{n}` to :math:`t_{n+1}` until
a terminal time :math:`T` is reached::

    # Output file
    file = File("output.pvd", "compressed")

    # Step in time
    t = 0.0
    T = 50*dt
    while (t < T):
        t += dt
        u0.vector()[:] = u.vector()
        solver.solve(problem, u.vector())
        file << (u.split()[0], t)

The string ``"compressed"`` indicates that the output data should be
compressed to reduce the file size. Within the time stepping loop, the
solution vector associated with ``u`` is copied to ``u0`` at the
beginning of each time step, and the nonlinear problem is solved by
calling :py:func:`solver.solve(problem, u.vector())
<dolfin.cpp.NewtonSolver.solve>`, with the new solution vector
returned in :py:func:`u.vector() <dolfin.cpp.Function.vector>`. The
``c`` component of the solution (the first component of ``u``) is then
written to file at every time step.
