burgers.pyΒΆ

In this script we solve the Burgers equation on a 1D or 2D periodic dommain, starting from a centered Gaussian and convecting in the positive direction of the first coordinate.

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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), number of dimensions, polynomial degree, time scale, Newton tolerance and the stopping time.

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def main(nelems: 'number of elements' = 20,
         ndims: 'spatial dimension' = 1,
         degree: 'polynomial degree' = 1,
         timescale: 'time scale (timestep=timescale/nelems)' = .5,
         newtontol: 'solver tolerance' = 1e-5,
         endtime: 'end time' = numpy.inf):

  domain, geom = nutils.mesh.rectilinear([numpy.linspace(0,1,nelems+1)]*ndims, periodic=range(ndims))

  ns = nutils.function.Namespace()
  ns.x = geom
  ns.basis = domain.basis('discont', degree=degree)
  ns.u = 'basis_n ?lhs_n'
  ns.f = '.5 u^2'
  ns.C = 1

  res = domain.integral('-basis_n,0 f d:x' @ ns, degree=5)
  res += domain.interfaces.integral('-[basis_n] n_0 ({f} - .5 C [u] n_0) d:x' @ ns, degree=degree*2)
  inertia = domain.integral('basis_n u d:x' @ ns, degree=5)

  sqr = domain.integral('(u - exp(-?y_i ?y_i)(y_i = 5 (x_i - 0.5_i)))^2 d:x' @ ns, degree=5)
  lhs0 = nutils.solver.optimize('lhs', sqr)

  timestep = timescale/nelems
  bezier = domain.sample('bezier', 7)
  with nutils.log.iter.plain('timestep', nutils.solver.impliciteuler('lhs', res, inertia, timestep=timestep, lhs0=lhs0, newtontol=newtontol)) as steps:
    for itime, lhs in enumerate(steps):
      x, u = bezier.eval(['x_i', 'u'] @ ns, lhs=lhs)
      nutils.export.triplot('solution.png', x, u, tri=bezier.tri, hull=bezier.hull, clim=(0,1))
      if itime * timestep >= endtime:
        break

  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. For example, to simulate until 0.5 seconds run python3 burgers.py endtime=0.5 (view log).

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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):

  @nutils.testing.requires('matplotlib')
  def test_1d_p1(self):
    lhs = main(ndims=1, nelems=10, timescale=.1, degree=1, endtime=.01)
    self.assertAlmostEqual64(lhs, '''
      eNrbocann6u3yqjTyMLUwfSw2TWzKPNM8+9mH8wyTMNNZxptMirW49ffpwYAI6cOVA==''')

  @nutils.testing.requires('matplotlib')
  def test_1d_p2(self):
    lhs = main(ndims=1, nelems=10, timescale=.1, degree=2, endtime=.01)
    self.assertAlmostEqual64(lhs, '''
      eNrr0c7SrtWfrD/d4JHRE6Ofxj6mnqaKZofNDpjZmQeYB5pHmL8we23mb5ZvWmjKY/LV6KPRFIMZ+o36
      8dp92gCxZxZG''')

  @nutils.testing.requires('matplotlib')
  def test_2d_p1(self):
    lhs = main(ndims=2, nelems=4, timescale=.1, degree=1, endtime=.01)
    self.assertAlmostEqual64(lhs, '''
      eNoNyKENhEAQRuGEQsCv2SEzyQZHDbRACdsDJNsBjqBxSBxBHIgJ9xsqQJ1Drro1L1/eYBZceGz8njrR
      yacm8UQLBvPYCw1airpyUVYSJLhKijK4IC01WDnqqxvX8OTl427aU73sctPGr3qqceBnRzOjo0xy9JpJ
      R73m6R6YMZo/Q+FCLQ==''')