integral over embedded, straight line

[joostvanzwieten]

Given a topology and two points, x0 and x1, inside the topology, how can I compute an integral over a straight line between x0 and x1?

[joostvanzwieten]

If the topology is 2d you can create a levelset that vanishes at the line through x0 and x1, trim the topology and take the trim boundary. Example for Nutils v6 and up:

import numpy
from nutils import function
from nutils.topology import Topology

def make_line_2d(topo: Topology,
                 x: function.Array,
                 x0: numpy.ndarray,
                 x1: numpy.ndarray,
                 end_at_points=True,
                 maxrefine: int = 0) -> Topology:
    '''Create a line topology through the given points.

    By default the line topology ends in the given points. Setting
    ``end_at_points`` to false lets the line extend to the bounary of the
    topology.

    Parameters
    ----------
    topo : :class:`nutils.topology.Topology`
        The 2D topology to make a line in.
    x : :class:`nutils.function.Array`
        The geometry.
    x0 : :class:`numpy.ndarray`
        Let the line go through this point.
    x1 : :class:`numpy.ndarray`
        Let the line go through this point as well.
    end_at_points : :class:`bool`
        If true the line topology ends at the given points, otherwise the line
        extends to the boundary of ``topo``.
    maxrefine : :class:`int`
        Number of times to refine. See :meth:`nutils.topology.Topology.trim`

    Example
    -------

    >>> import numpy
    >>> from nutils import function, mesh
    >>> topo, geom = mesh.rectilinear([5, 5])
    >>> line = make_line_2d(topo, geom, numpy.array([0.15, 0.35]), numpy.array([3.15, 4.35]))
    >>> line.integrate(function.J(geom), degree=0).round(3)
    5.0
    '''

    t = x1 - x0
    n = numpy.array([-t[1], t[0]])
    topo = topo.trim((x - x0) @ n, maxrefine=maxrefine, name='line')
    if end_at_points:
        topo = topo.trim(t @ t - abs((x1 + x0 - 2 * x) @ t), maxrefine=maxrefine)
    return topo.boundary['line']

[Siamakpk]

Thanks, Joost. What of the topology is 1D?

[joostvanzwieten]

You can use trim as well. Given two scalar points x0 and x1 and scalar geometry x you can do the following:

line = topo.trim(abs(x1 - x0) - abs(x1 + x0 - 2 * x), maxrefine=0)

[mcurti]

You could also discretize the line with Gaussian quadratures, locate the points with locate and use the quadrature weights for integration.