Sampling the solution over a line

Let us revisit our simple 3d example from the introduction

from matplotlib import pyplot
import numpy
  import pygmsh
  geom = pygmsh.built_in.Geometry()
  poly = geom.add_polygon([
      [ 0.0,  0.5, 0.0], [-0.1,  0.1, 0.0], [-0.5,  0.0, 0.0],
      [-0.1, -0.1, 0.0], [ 0.0, -0.5, 0.0], [ 0.1, -0.1, 0.0],
      [ 0.5,  0.0, 0.0], [ 0.1,  0.1, 0.0] ], lcar=0.05)
  axis = [0, 0, 1]
  geom.extrude( poly, translation_axis=axis, rotation_axis=axis,
      point_on_axis=[0, 0, 0], angle=2.0 / 6.0 * numpy.pi)
  mesh = pygmsh.generate_mesh(geom, verbose=False)
  points, cells = mesh.points, mesh.cells
  domain3d = {"vertices":points, "simplices":cells["tetra"]}
except ImportError: # pygmsh not installed - use a simple cartesian domain
  from dune.grid import cartesianDomain
  domain3d = cartesianDomain([-0.25,-0.25,0],[0.25,0.25,1],[30,30,60])

from dune.alugrid import aluSimplexGrid as leafGridView3d
gridView3d  = leafGridView3d(domain3d)

As before we solve a simple Laplace problem

from import lagrange as solutionSpace
from dune.fem.scheme import galerkin as solutionScheme
from ufl import TrialFunction, TestFunction, SpatialCoordinate, dot, grad, dx, conditional, sqrt

space3d = solutionSpace(gridView3d, order=1)
u = TrialFunction(space3d)
v = TestFunction(space3d)
x = SpatialCoordinate(space3d)
scheme3d = solutionScheme((dot(grad(u),grad(v))+u*v)*dx ==
uh3d = space3d.interpolate([0],name="solution")
info = scheme3d.solve(target=uh3d)

Instead of plotting this using paraview we want to only study the solution along a single line. This requires findings points \(x_i = x_0+\frac{i}{N}(x1-x0)\) for \(i=0,\dots,N\) within the unstructured grid. This would be expensive to compute on the Python so we implement this algorithm in C++ using the LineSegmentSampler class available in Dune-Fem. The resulting algorithm returns a pair of two lists with coordinates \(x_i\) and the values of the grid function at these points:

#include <vector>
#include <utility>
#include <dune/fem/misc/linesegmentsampler.hh>

template <class GF, class DT>
std::pair<std::vector<DT>, std::vector<typename GF::RangeType>>
sample(const GF &gf, DT &start, DT &end, int n)
  Dune::Fem::LineSegmentSampler<typename GF::GridPartType> sampler(gf.gridPart(),start,end);
  std::vector<DT> coords(n);
  std::vector<typename GF::RangeType> values(n);
  return std::make_pair(coords,values);
import dune.generator.algorithm as algorithm
from dune.common import FieldVector
x0, x1 = FieldVector([0,0,0]), FieldVector([0,0,1])
p,v ='sample', 'utility.hh', uh3d, x0, x1, 100)
x,y = numpy.zeros(len(p)), numpy.zeros(len(p))
length = (x1-x0).two_norm
for i in range(len(x)):
    x[i] = (p[i]-x0).two_norm / length
    y[i] = v[i][0]

Note: the coordinates returned are always in the interval \([0,1]\) so if physical coordinates are required, they need to be rescaled. Also, function values returned by the sample function are always of a FieldVector type, so that even for a scalar example a v[i] is a vector of dimension one, so that y[i]=v[i][0] has to be used.

A mentioned above any grid function can be passed in as argument to the sample function. So for example plotting \(|\nabla u_h|\) is straight forward using the corresponding ufl expression. Since in this case automatic conversion from the ufl expression (available for example in the plotting function) to a grid function, we need to do this explicitly:

from dune.ufl import expression2GF
absGrad = expression2GF(gridView3d, sqrt(dot(grad(uh3d),grad(uh3d))), 2 )
p,v ='sample', 'utility.hh', absGrad, x0, x1, 100)
for i in range(len(x)):
    y[i] = v[i][0]

Similar we can plot both partial derivatives of the solution over the given line:

from dune.ufl import expression2GF
absGrad = expression2GF(gridView3d, grad(uh3d), 2 )
p,v ='sample', 'utility.hh', absGrad, x0, x1, 100)
dx,dy = numpy.zeros(len(p)), numpy.zeros(len(p))
for i in range(len(x)):
    dx[i] = v[i][0]
    dy[i] = v[i][1]

Remark: plotting over a line has been included as a utility function so can easily be called as shown below

from dune.fem.utility import lineSample
x,y = lineSample(uh3d,[0,0,0],[0,0,1],100)