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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import nutils, numpy

The main function defines the parameter space for the script. Configurable parameters are the mesh density (in number of elements along an edge), element type (square, triangle, or mixed), type of basis function (std or spline, with availability depending on element type), polynomial degree, the epsilon parameter, contactangle, timestep, stop criterion, random seed, and a boolean flag for making the domain circular as opposed to a unit square.

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def main(nelems: 'number of elements' = 20,
         etype: 'type of elements (square/triangle/mixed)' = 'square',
         btype: 'type of basis function (std/spline)' = 'std',
         degree: 'polynomial degree' = 2,
         epsilon: 'epsilon, 0 for automatic (based on nelems)' = 0,
         contactangle: 'wall contact angle (degrees)' = 90,
         timestep: 'time step' = .01,
         mtol: 'stop when chemical potential is peak to peak below threshold' = .01,
         seed: 'random seed' = 0,
         circle: 'select circular domain' = False):

  mineps = 1./nelems
  if not epsilon:
    nutils.log.info('setting epsilon={}'.format(mineps))
    epsilon = mineps
  elif epsilon < mineps:
    nutils.log.warning('epsilon under crititical threshold: {} < {}'.format(epsilon, mineps))

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

  ns = nutils.function.Namespace()
  if not circle:
    ns.x = geom
  else:
    ns.xi = (geom-.5) * (.5*numpy.pi)
    ns.x_i = '<sin(xi_0) cos(xi_1), cos(xi_0) sin(xi_1)>_i / sqrt(2)'
  ns.epsilon = epsilon
  ns.ewall = .5 * numpy.cos(contactangle * numpy.pi / 180)
  ns.cbasis, ns.mbasis = nutils.function.chain([domain.basis('std', degree=degree)] * 2)
  ns.c = 'cbasis_n ?lhs_n'
  ns.c0 = 'cbasis_n ?lhs0_n'
  ns.m = 'mbasis_n ?lhs_n'
  ns.f = '(6 c0 - 2 c0^3 - 4 c) / epsilon^2' # convex/concave splitting of double well potential derivative

  res = domain.integral('(epsilon^2 mbasis_n,k m_,k + cbasis_n,k c_,k) d:x' @ ns, degree=7)
  res -= domain.integral('cbasis_n (m + f) d:x' @ ns, degree=7)
  res += domain.boundary.integral('cbasis_n ewall d:x' @ ns, degree=7)
  inertia = domain.integral('mbasis_n c d:x' @ ns, degree=7)

  energy = dict( # energy breakdown
    mixture = domain.integral('(c^2 - 1)^2 d:x / 2 epsilon^2' @ ns, degree=4),
    interfaces = domain.integral('.5 c_,k c_,k d:x' @ ns, degree=4),
    wall = domain.boundary.integral('(abs(ewall) + ewall c) d:x' @ ns, degree=4))

  numpy.random.seed(seed)
  lhs0 = numpy.random.normal(0, .5, ns.cbasis.shape) # initial condition

  with nutils.log.iter.plain('timestep', nutils.solver.impliciteuler('lhs', target0='lhs0', residual=res, inertia=inertia, timestep=timestep, lhs0=lhs0)) as steps:
    for lhs in steps:

      E = nutils.sample.eval_integrals(*energy.values(), lhs=lhs)
      nutils.log.user('energy: {:.3f} ({})'.format(sum(E), ', '.join('{:.0f}% {}'.format(100*e/sum(E), n) for e, n in sorted(zip(E, energy), reverse=True))))

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

      if numpy.ptp(m) < mtol:
        break

  return lhs0, 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__':
  nutils.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(nutils.testing.TestCase):

  def _checkrand(self, lhs0):
    with self.subTest('initial condition'): self.assertAlmostEqual64(lhs0, '''
      eNoBxAA7/xM3LjTtNYs3MDcUyt41uc14zjo0LzKzNm812jFhNNMzwDYgzbMzV8o0yCM1rzWeypE3Tcnx
      L07NzTa4NlMyETREyrPIGMxYMl82VDbjy1/M8clZyf3IRjday6XLmMl6NRnJDs1Ayh00WMu1yQHRUDSs
      MKIz7MoEzM/KCMxwyvjIlzLQyxTJdjQ5yjEwWjX3MTk2n8kwNMbKTsoay1DMWDC8ycM1eTQyyb42NzdK
      NmLN5skSNs/LXDbnMuw19DNKNREtGTfui1ut''')

  @nutils.testing.requires('matplotlib')
  def test_square(self):
    lhs0, lhs = main(nelems=3, timestep=1, mtol=.1)
    self._checkrand(lhs0)
    with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
      eNqbZTbHzMHsiGmpCd9V1gszzWaZ2ZjtMQ01eXV+xbk0szSgzAaTDxdNTkue1jbTMpM15TJqP/335PeT
      100vmyqYaJ3tPNV1svNknmmKqYJR+On3J01Pmp9MMY0y/WIYCOSZn7Q82XCi8UTXiSkn5pxYBISovJYT
      rSd6T0wD8xae6ATCCSemn5gLlusFwiknZp9YcGIpEE4Ewhkn5p1YfGIFEKLyAN6wcSE=''')

  @nutils.testing.requires('matplotlib')
  def test_contactangle(self):
    lhs0, lhs = main(nelems=3, timestep=1, mtol=.1, contactangle=45)
    self._checkrand(lhs0)
    with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
      eNqzNsszkzZbbfrdOOus6Jlss5lmPmbPTQtNtp6be8bZrNTss6mW6SMDv9OnTokDZRpMbxl7nNE89fTk
      ItNHpl0mT8+fOzX3ZP7J3yb+ph1G206zn7I+KXWyyOSeibK+1ulzJyVP/joRZhJp0m6yyeSyyXsgDAfy
      2kw2mlw0eWvyxiTLJNtkgslmk3Mmz4CwzqTeZLbJNpOzJo+AcIrJVJO1JkdMbpi8BsLlJitM9gHNeGLy
      2eQLkLfSZL/JFZOnJl+BEAAJrlyi''')

  @nutils.testing.requires('matplotlib')
  def test_mixedcircle(self):
    lhs0, lhs = main(nelems=3, timestep=1, mtol=.1, circle=True, etype='mixed')
    self._checkrand(lhs0)
    with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
      eNrTM31uImDqY1puGmwia1prssNY37TERNM01eSOkYuJlck6Q1ED9TP9px+fOmq82FjtfKFJiM6CK70m
      BsZixmUXgk9XnMo7VX6661zL+cZz58+ln0s6e/PM7DOvjDTOvTz97tS8c6xn9pzYemLHiQMn9p9YDyS3
      nth4YteJbUCRHUByO5DcfGLDieUnlpyYA2RtP7HpxJ4T64Aih8Bwz4k1QPF5QJ3rgap3ntgCVAHRe+bE
      biBr5YmDQBMBKJ13Eg==''')