cahnhilliard.pyΒΆ

In this script we solve the Cahn-Hiilliard equation, which models the unmixing of two phases under the effect of surface tension.

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from nutils import mesh, function, solver, sample, export, cli, testing
import numpy, treelog, itertools, enum, typing

class stab(enum.Enum):
  none = '0' # for educational purposes only
  linear = '.5 dc^2 (6 - 6 c^2 + 8 c dc - 3 dc^2) / epsilon^2'
  optimal = '.5 dc^2 (1 - dc^2 / 12) / epsilon^2'

def main(nelems:int, etype:str, btype:str, degree:int, epsilon:typing.Optional[float],
         contactangle:float, timestep:float, mtol:float, seed:int, circle:bool, stab:stab):
  '''
  Cahn-Hilliard equation on a unit square/circle.

  .. arguments::

     nelems [20]
       Number of elements along domain edge.
     etype [square]
       Type of elements (square/triangle/mixed).
     btype [std]
       Type of basis function (std/spline), with availability depending on the
       configured element type.
     degree [2]
       Polynomial degree.
     epsilon []
       Interface thickness; defaults to an automatic value based on the
       configured mesh density if left unspecified.
     contactangle [90]
       Wall contact angle in degrees.
     timestep [.01]
       Time step.
     mtol [.01]
       Threshold value for chemical potential peak to peak difference, used as
       a stop criterion.
     seed [0]
       Random seed for the initial condition.
     circle [no]
       Select circular domain as opposed to a unit square.
     stab [linear]
       Stabilization method (linear/optimal/none).
  '''

  mineps = 1./nelems
  if epsilon is None:
    treelog.info('setting epsilon={}'.format(mineps))
    epsilon = mineps
  elif epsilon < mineps:
    treelog.warning('epsilon under crititical threshold: {} < {}'.format(epsilon, mineps))

  domain, geom = mesh.unitsquare(nelems, etype)
  bezier = domain.sample('bezier', 5) # sample for plotting

  ns = function.Namespace()
  if not circle:
    ns.x = geom
  else:
    angle = (geom-.5) * (numpy.pi/2)
    ns.x = function.sin(angle) * function.cos(angle)[[1,0]] / numpy.sqrt(2)
  ns.epsilon = epsilon
  ns.ewall = .5 * numpy.cos(contactangle * numpy.pi / 180)
  ns.cbasis, ns.mbasis = function.chain([domain.basis('std', degree=degree)] * 2)
  ns.c = 'cbasis_n ?lhs_n'
  ns.dc = 'cbasis_n (?lhs_n - ?lhs0_n)'
  ns.m = 'mbasis_n ?lhs_n'
  ns.F = '.5 (c^2 - 1)^2 / epsilon^2'
  ns.dF = stab.value
  ns.dt = timestep

  nrg_mix = domain.integral('F d:x' @ ns, degree=7)
  nrg_iface = domain.integral('.5 c_,k c_,k d:x' @ ns, degree=7)
  nrg_wall = domain.boundary.integral('(abs(ewall) + c ewall) d:x' @ ns, degree=7)
  nrg = nrg_mix + nrg_iface + nrg_wall + domain.integral('(dF - m dc - .5 dt epsilon^2 m_,k m_,k) d:x' @ ns, degree=7)

  numpy.random.seed(seed)
  lhs = numpy.random.normal(0, .5, ns.cbasis.shape) # initial condition

  with treelog.iter.plain('timestep', itertools.count()) as steps:
   for istep in steps:

    E = sample.eval_integrals(nrg_mix, nrg_iface, nrg_wall, lhs=lhs)
    treelog.user('energy: {0:.3f} ({1[0]:.0f}% mixture, {1[1]:.0f}% interface, {1[2]:.0f}% wall)'.format(sum(E), 100*numpy.array(E)/sum(E)))

    x, c, m = bezier.eval(['x_i', 'c', 'm'] @ ns, lhs=lhs)
    export.triplot('phase.png', x, c, tri=bezier.tri, clim=(-1,1))
    export.triplot('chempot.png', x, m, tri=bezier.tri)

    if numpy.ptp(m) < mtol:
      break

    lhs = solver.optimize('lhs', nrg, arguments=dict(lhs0=lhs), lhs0=lhs, newtontol=1e-10)

  return lhs

If the script is executed (as opposed to imported), nutils.cli.run() calls the main function with arguments provided from the command line.

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if __name__ == '__main__':
  cli.run(main)

Once a simulation is developed and tested, it is good practice to save a few strategic return values for regression testing. The nutils.testing module, which builds on the standard unittest framework, facilitates this by providing nutils.testing.TestCase.assertAlmostEqual64() for the embedding of desired results as compressed base64 data.

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class test(testing.TestCase):

  @testing.requires('matplotlib')
  def test_initial(self):
    lhs = main(nelems=3, etype='square', btype='std', degree=2, epsilon=None, contactangle=90, timestep=1, mtol=float('inf'), seed=0, circle=False, stab=stab.linear)
    self.assertAlmostEqual64(lhs, '''
      eNoBxAA7/xM3LjTtNYs3MDcUyt41uc14zjo0LzKzNm812jFhNNMzwDYgzbMzV8o0yCM1rzWeypE3Tcnx
      L07NzTa4NlMyETREyrPIGMxYMl82VDbjy1/M8clZyf3IRjday6XLmMl6NRnJDs1Ayh00WMu1yQHRUDSs
      MKIz7MoEzM/KCMxwyvjIlzLQyxTJdjQ5yjEwWjX3MTk2n8kwNMbKTsoay1DMWDC8ycM1eTQyyb42NzdK
      NmLN5skSNs/LXDbnMuw19DNKNREtGTfui1ut''')

  @testing.requires('matplotlib')
  def test_square(self):
    lhs = main(nelems=3, etype='square', btype='std', degree=2, epsilon=None, contactangle=90, timestep=1, mtol=.1, seed=0, circle=False, stab=stab.linear)
    self.assertAlmostEqual64(lhs, '''
      eNqbZTbHzMHsiGmpCd9V1gszzWaZ2ZjtMQ01eXV+xbk0szSgzAaTDxdNTkue1jbTMpM15TJqP/335PeT
      100vmyqYaJ3tPNV1svNknmmKqYJR+On3J01Pmp9MMY0y/WIYCOSZn7Q82XCi8UTXiSkn5pxYBISovJYT
      rSd6T0wD8xae6ATCCSemn5gLlusFwiknZp9YcGIpEE4Ewhkn5p1YfGIFEKLyAN6wcSE=''')

  @testing.requires('matplotlib')
  def test_contactangle(self):
    lhs = main(nelems=3, etype='square', btype='std', degree=2, epsilon=None, contactangle=45, timestep=1, mtol=.1, seed=0, circle=False, stab=stab.linear)
    self.assertAlmostEqual64(lhs, '''
      eNqzNsszkzZbbfrdOOus6Jlss5lmPmbPTQtNtp6be8bZrNTss6mW6SMDv9OnTokDZRpMbxl7nNE89fTk
      ItNHpl0mT8+fOzX3ZP7J3yb+ph1G206zn7I+KXWyyOSeibK+1ulzJyVP/joRZhJp0m6yyeSyyXsgDAfy
      2kw2mlw0eWvyxiTLJNtkgslmk3Mmz4CwzqTeZLbJNpOzJo+AcIrJVJO1JkdMbpi8BsLlJitM9gHNeGLy
      2eQLkLfSZL/JFZOnJl+BEAAJrlyi''')

  @testing.requires('matplotlib')
  def test_mixedcircle(self):
    lhs = main(nelems=3, etype='mixed', btype='std', degree=2, epsilon=None, contactangle=90, timestep=1, mtol=.1, seed=0, circle=True, stab=stab.linear)
    self.assertAlmostEqual64(lhs, '''
      eNrTM31uImDqY1puGmwia1prssNY37TERNM01eSOkYuJlck6Q1ED9TP9px+fOmq82FjtfKFJiM6CK70m
      BsZixmUXgk9XnMo7VX6661zL+cZz58+ln0s6e/PM7DOvjDTOvTz97tS8c6xn9pzYemLHiQMn9p9YDyS3
      nth4YteJbUCRHUByO5DcfGLDieUnlpyYA2RtP7HpxJ4T64Aih8Bwz4k1QPF5QJ3rgap3ntgCVAHRe+bE
      biBr5YmDQBMBKJ13Eg==''')