Interpolation and IO

This demo (demo_interpolation-io.py) shows the interpolation of functions into vector-element \(H(\mathrm{curl})\) finite element spaces, and the interpolation of these special finite elements in discontinuous Lagrange spaces for artifact-free visualisation.

import numpy as np

from dolfinx import default_scalar_type, plot
from dolfinx.fem import Function, functionspace
from dolfinx.mesh import CellType, create_rectangle, locate_entities

from mpi4py import MPI

Create a mesh. For later in the demo we need to ensure that a boundary between cells is located at \(x_0=0.5\).

msh = create_rectangle(MPI.COMM_WORLD, ((0.0, 0.0), (1.0, 1.0)), (16, 16), CellType.triangle)

Create a Nédélec function space and finite element Function

V = functionspace(msh, ("Nedelec 1st kind H(curl)", 1))
u = Function(V, dtype=default_scalar_type)

Find cells with all vertices (0) \(x_0 <= 0.5\) or (1) \(x_0 >= 0.5\):

tdim = msh.topology.dim
cells0 = locate_entities(msh, tdim, lambda x: x[0] <= 0.5)
cells1 = locate_entities(msh, tdim, lambda x: x[0] >= 0.5)

Interpolate in the Nédélec/H(curl) space a vector-valued expression f, where \(f \cdot n\) is discontinuous at \(x_0 = 0.5\) and \(f \cdot e\) is continuous.

u.interpolate(lambda x: np.vstack((x[0], x[1])), cells0)
u.interpolate(lambda x: np.vstack((x[0] + 1, x[1])), cells1)

Create a vector-valued discontinuous Lagrange space and function, and interpolate the \(H({\rm curl})\) function u

gdim = msh.geometry.dim
V0 = functionspace(msh, ("Discontinuous Lagrange", 1, (gdim,)))
u0 = Function(V0, dtype=default_scalar_type)
u0.interpolate(u)

We save the interpolated function u0 in VTX format. When visualising the field, at \(x_0 = 0.5\) the \(x_0\)-component should appear discontinuous and the \(x_1\)-component should appear continuous.

try:
    from dolfinx.io import VTXWriter
    with VTXWriter(msh.comm, "output_nedelec.bp", u0, "bp4") as f:
        f.write(0.0)
except ImportError:
    print("ADIOS2 required for VTX output")

Plot the functions

try:
    import pyvista
    cells, types, x = plot.vtk_mesh(V0)
    grid = pyvista.UnstructuredGrid(cells, types, x)
    values = np.zeros((x.shape[0], 3), dtype=np.float64)
    values[:, :msh.topology.dim] = u0.x.array.reshape(x.shape[0], msh.topology.dim).real
    grid.point_data["u"] = values

    pl = pyvista.Plotter(shape=(2, 2))

    pl.subplot(0, 0)
    pl.add_text("magnitude", font_size=12, color="black", position="upper_edge")
    pl.add_mesh(grid.copy(), show_edges=True)

    pl.subplot(0, 1)
    glyphs = grid.glyph(orient="u", factor=0.08)
    pl.add_text("vector glyphs", font_size=12, color="black", position="upper_edge")
    pl.add_mesh(glyphs, show_scalar_bar=False)
    pl.add_mesh(grid.copy(), style="wireframe", line_width=2, color="black")

    pl.subplot(1, 0)
    pl.add_text("x-component", font_size=12, color="black", position="upper_edge")
    pl.add_mesh(grid.copy(), component=0, show_edges=True)

    pl.subplot(1, 1)
    pl.add_text("y-component", font_size=12, color="black", position="upper_edge")
    pl.add_mesh(grid.copy(), component=1, show_edges=True)

    pl.view_xy()
    pl.link_views()

    # If pyvista environment variable is set to off-screen (static)
    # plotting save png
    if pyvista.OFF_SCREEN:
        pyvista.start_xvfb(wait=0.1)
        pl.screenshot("uh.png")
    else:
        pl.show()
except ModuleNotFoundError:
    print("pyvista is required to visualise the solution")