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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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), polynomial degree, and Reynolds number.

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def main(nelems: 'number of elements' = 12,
         degree: 'polynomial degree for velocity' = 2,
         reynolds: 'reynolds number' = 1000.):

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

  ns = nutils.function.Namespace()
  ns.x = geom
  ns.Re = reynolds
  ns.uxbasis, ns.uybasis, ns.pbasis, ns.lbasis = nutils.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 nutils.log.context('stokes'):
    lhs0 = nutils.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 nutils.log.context('navierstokes'):
    lhs1 = nutils.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
  nutils.log.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'] = nutils.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 nutils.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__':
  nutils.cli.run(main)

Once a simulation is developed and tested, it is good practice to save a few strategicly chosen return values for routine regression testing. Here we use the standard unittest framework, with nutils.numeric.assert_allclose64() facilitating the embedding of desired results as compressed base64 data.

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class test(nutils.testing.TestCase):

  @nutils.testing.requires('matplotlib')
  def test_p1(self):
    lhs0, lhs1 = main(nelems=3, reynolds=100, degree=2)
    nutils.numeric.assert_allclose64(lhs0, 'eNpTvPBI3/o0t1mzds/pltM65opQ/n196QvcZh4XO03MTHbolZ'
      '8+dVrxwlP9rycVL03Xjbm45tQfrZc37M/LGLBcFVc/aPDk/H3dzEtL9EJMGRgAJt4mPA==')
    nutils.numeric.assert_allclose64(lhs1, 'eNoBUgCt/6nOuTGJy4M1SCzJy4zLCjcsLk3PCst/Nlcx9M2DNe'
      'DPgDR+NB7UG8wVzSwuPc6ByezUQiudMKTL/y4AL73NLS6jLUov8s4zzXoscdMJMSo2AABO+yTF')

  @nutils.testing.requires('matplotlib')
  def test_p2(self):
    lhs0, lhs1 = main(nelems=3, reynolds=100, degree=3)
    nutils.numeric.assert_allclose64(lhs0, 'eNp7ZmB71sY46VSq2dLzludvnMo20jFHsJ7BZaXObzbedDrVbJ'
      'nBjPM1ZkuNGaAg6nyGQcvJ6DPPDHzP+JnMPsltwKl1/DyrYcPJUxf0LuXqvDkzzYgBDsz0L+lOvixinH'
      'X26/nvVy0Nfp9rMGNgAADUrDbX')
    nutils.numeric.assert_allclose64(lhs1, 'eNoBhAB7/3Axm8zRM23KHDbJzyrMAs7DzOY2yM/vLvfJ8TQ/N8'
      'AvSc5FMkjKwTaQzlo0K8scNuwwLDKfNWQzcCLOzCs1jTEA0FcxA8kLzcAvU81jMz/JVTELMUjOLDL+ye'
      'MsaS6lLkLOajM9LDgwWNBzzOvOMTBCMHnXnDHFzcDTYDCgKo0vLzcAACOlOuU=')