Sampling

Having obtained the coefficient vector that solves the Laplace problem, we are now interested in visualizing the function it represents. Nutils does not provide its own post processing functionality, leaving that up to the preference of the user. It does, however, facilitate it, by allowing nutils.function.Array functions to be evaluated in samples. Bundling function values and a notion of connectivity, these form a bridge between Nutils' world of functions and the discrete realms of matplotlib, VTK, etc.

The Topology.sample(method, ...) method generates a collection of points on the nutils.topology.Topology, according to method. The 'bezier' method generates equidistant points per element, including the element vertices. The number of points per element per dimension is controlled by the second argument of Topology.sample(). An example:

bezier = topo.sample('bezier', 2)

The resulting nutils.sample.Sample object can be used to evaluate nutils.function.Array functions via the Sample.eval(func) method. To evaluate the geometry ns.x write

x = bezier.eval('x_0' @ ns)
x
# array([0.  , 0.25, 0.25, 0.5 , 0.5 , 0.75, 0.75, 1.  ])±1e-15

The first axis of the returned numpy.ndarray represents the collection of points. To reorder this into a sequence of lines in 1D, a triangulation in 2D or in general a sequence of simplices, use the Sample.tri attribute:

x.take(bezier.tri, 0)
# array([[0.  , 0.25],
#        [0.25, 0.5 ],
#        [0.5 , 0.75],
#        [0.75, 1.  ]])±1e-15

Now, the first axis represents the simplices and the second axis the vertices of the simplices.

If an nutils.function.Array function has arguments, those arguments must be specified by keyword arguments to Sample.eval(). For example, to evaluate ns.u with argument lhs replaced by solution vector lhs, obtained using nutils.solver.solve_linear above, write

u = bezier.eval('u' @ ns, lhs=lhs)
u
# array([0.  , 0.25, 0.25, 0.5 , 0.5 , 0.75, 0.75, 1.  ])±1e-15

We can now plot the sampled geometry x and solution u using matplotlib_, plotting each line in Sample.tri with a different color:

>>> plt.plot(x.take(bezier.tri.T, 0), u.take(bezier.tri.T, 0))

Recall that we have imported matplotlib.pyplot as plt above. The plt.plot() function takes an array of x-values and and array of y-values, both with the first axis representing vertices and the second representing separate lines, hence the transpose of bezier.tri.

The plt.plot() function also supports plotting lines with discontinuities, which are represented by nan values. We can use this to plot the solution as a single, but possibly discontinuous line. The function numpy.insert can be used to prepare a suitable array. An example:

nanjoin = lambda array, tri: numpy.insert(array.take(tri.flat, 0).astype(float),
    slice(tri.shape[1], tri.size, tri.shape[1]), numpy.nan, axis=0)
nanjoin(x, bezier.tri)
# array([0.  , 0.25,  nan, 0.25, 0.5 ,  nan, 0.5 , 0.75,  nan, 0.75, 1.  ])±1e-15
plt.plot(nanjoin(x, bezier.tri), nanjoin(u, bezier.tri))

Note the difference in colors between the last two plots.