drivencavity-compatible.py

In this script we solve the lid driven cavity problem for stationary Stokes and Navier-Stokes flow. That is, a unit square domain, with no-slip left, bottom and right boundaries and a top boundary that is moving at unit velocity in positive x-direction.

The script is identical to drivencavity.py except that it uses the Raviart-Thomas discretization providing compatible velocity and pressure spaces resulting in a pointwise divergence-free velocity field.

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from nutils import mesh, function, solver, export, cli, testing
import numpy, treelog

def main(nelems:int, degree:int, reynolds:float):
  '''
  Driven cavity benchmark problem using compatible spaces.

  .. arguments::

     nelems [12]
       Number of elements along edge.
     degree [2]
       Polynomial degree for velocity; the pressure space is one degree less.
     reynolds [1000]
       Reynolds number, taking the domain size as characteristic length.
  '''

  verts = numpy.linspace(0, 1, nelems+1)
  domain, geom = mesh.rectilinear([verts, verts])

  ns = function.Namespace()
  ns.x = geom
  ns.Re = reynolds
  ns.uxbasis, ns.uybasis, ns.pbasis, ns.lbasis = function.chain([
    domain.basis('spline', degree=(degree,degree-1), removedofs=((0,-1),None)),
    domain.basis('spline', degree=(degree-1,degree), removedofs=(None,(0,-1))),
    domain.basis('spline', degree=degree-1),
    [1], # lagrange multiplier
  ])
  ns.ubasis_ni = '<uxbasis_n, uybasis_n>_i'
  ns.u_i = 'ubasis_ni ?lhs_n'
  ns.p = 'pbasis_n ?lhs_n'
  ns.l = 'lbasis_n ?lhs_n'
  ns.stress_ij = '(u_i,j + u_j,i) / Re - p δ_ij'
  ns.uwall = domain.boundary.indicator('top'), 0
  ns.N = 5 * degree * nelems # nietzsche constant

  res = domain.integral('(ubasis_ni,j stress_ij + pbasis_n (u_k,k + l) + lbasis_n p) d:x' @ ns, degree=2*degree)
  res += domain.boundary.integral('(N ubasis_ni - (ubasis_ni,j + ubasis_nj,i) n_j) (u_i - uwall_i) d:x / Re' @ ns, degree=2*degree)
  with treelog.context('stokes'):
    lhs0 = solver.solve_linear('lhs', res)
    postprocess(domain, ns, lhs=lhs0)

  res += domain.integral('ubasis_ni u_i,j u_j d:x' @ ns, degree=3*degree)
  with treelog.context('navierstokes'):
    lhs1 = solver.newton('lhs', res, lhs0=lhs0).solve(tol=1e-10)
    postprocess(domain, ns, lhs=lhs1)

  return lhs0, lhs1

Postprocessing in this script is separated so that it can be reused for the results of Stokes and Navier-Stokes, and because of the extra steps required for establishing streamlines.

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def postprocess(domain, ns, every=.05, spacing=.01, **arguments):

  div = domain.integral('(u_k,k)^2 d:x' @ ns, degree=1).eval(**arguments)**.5
  treelog.info('velocity divergence: {:.2e}'.format(div)) # confirm that velocity is pointwise divergence-free

  ns = ns.copy_() # copy namespace so that we don't modify the calling argument
  ns.streambasis = domain.basis('std', degree=2)[1:] # remove first dof to obtain non-singular system
  ns.stream = 'streambasis_n ?streamdofs_n' # stream function
  sqr = domain.integral('((u_0 - stream_,1)^2 + (u_1 + stream_,0)^2) d:x' @ ns, degree=4)
  arguments['streamdofs'] = solver.optimize('streamdofs', sqr, arguments=arguments) # compute streamlines

  bezier = domain.sample('bezier', 9)
  x, u, p, stream = bezier.eval(['x_i', 'sqrt(u_i u_i)', 'p', 'stream'] @ ns, **arguments)
  with export.mplfigure('flow.png') as fig: # plot velocity as field, pressure as contours, streamlines as dashed
    ax = fig.add_axes([.1,.1,.8,.8], yticks=[], aspect='equal')
    import matplotlib.collections
    ax.add_collection(matplotlib.collections.LineCollection(x[bezier.hull], colors='w', linewidths=.5, alpha=.2))
    ax.tricontour(x[:,0], x[:,1], bezier.tri, stream, 16, colors='k', linestyles='dotted', linewidths=.5, zorder=9)
    caxu = fig.add_axes([.1,.1,.03,.8], title='velocity')
    imu = ax.tripcolor(x[:,0], x[:,1], bezier.tri, u, shading='gouraud', cmap='jet')
    fig.colorbar(imu, cax=caxu)
    caxu.yaxis.set_ticks_position('left')
    caxp = fig.add_axes([.87,.1,.03,.8], title='pressure')
    imp = ax.tricontour(x[:,0], x[:,1], bezier.tri, p, 16, cmap='gray', linestyles='solid')
    fig.colorbar(imp, cax=caxp)

If the script is executed (as opposed to imported), nutils.cli.run() calls the main function with arguments provided from the command line. To keep with the default arguments simply run python3 drivencavity-compatible.py (view log).

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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_p1(self):
    lhs0, lhs1 = main(nelems=3, reynolds=100, degree=2)
    with self.subTest('stokes'): self.assertAlmostEqual64(lhs0, '''
      eNpTvPBI3/o0t1mzds/pltM65opQ/n196QvcZh4XO03MTHbolZ8+dVrxwlP9rycVL03Xjbm45tQfrZc3
      7M/LGLBcFVc/aPDk/H3dzEtL9EJMGRgAJt4mPA==''')
    with self.subTest('navier-stokes'): self.assertAlmostEqual64(lhs1, '''
      eNoBUgCt/6nOuTGJy4M1SCzJy4zLCjcsLk3PCst/Nlcx9M2DNeDPgDR+NB7UG8wVzSwuPc6ByezUQiud
      MKTL/y4AL73NLS6jLUov8s4zzXoscdMJMSo2AABO+yTF''')

  @testing.requires('matplotlib')
  def test_p2(self):
    lhs0, lhs1 = main(nelems=3, reynolds=100, degree=3)
    with self.subTest('stokes'): self.assertAlmostEqual64(lhs0, '''
      eNp7ZmB71sY46VSq2dLzludvnMo20jFHsJ7BZaXObzbedDrVbJnBjPM1ZkuNGaAg6nyGQcvJ6DPPDHzP
      +JnMPsltwKl1/DyrYcPJUxf0LuXqvDkzzYgBDsz0L+lOvixinHX26/nvVy0Nfp9rMGNgAADUrDbX''')
    with self.subTest('navier-stokes'): self.assertAlmostEqual64(lhs1, '''
      eNoBhAB7/3Axm8zRM23KHDbJzyrMAs7DzOY2yM/vLvfJ8TQ/N8AvSc5FMkjKwTaQzlo0K8scNuwwLDKf
      NWQzcCLOzCs1jTEA0FcxA8kLzcAvU81jMz/JVTELMUjOLDL+yeMsaS6lLkLOajM9LDgwWNBzzOvOMTBC
      MHnXnDHFzcDTYDCgKo0vLzcAACOlOuU=''')