# 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.

 12 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.

 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 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 = '_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.

 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 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).

 90 91 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.

 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 class test(nutils.testing.TestCase): @nutils.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''') @nutils.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=''')