2D FEM Using Firedrake¶

Copyright (C) 2020 Andreas Kloeckner

MIT License Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
In [1]:
from firedrake import *
import numpy as np
import numpy.linalg as la
import firedrake.mesh as fd_mesh
import matplotlib.pyplot as plt

firedrake:WARNING OMP_NUM_THREADS is not set or is set to a value greater than 1, we suggest setting OMP_NUM_THREADS=1 to improve performance

Mesh the Domain¶

This uses meshpy, which under the hood uses Triangle.

pip install meshpy to install.

NB: Triangle is not open-source software. If you are looking for a quality mesher that is open-source (but a bit more complex to use), look at Gmsh.

In [3]:
import meshpy.triangle as triangle

def round_trip_connect(start, end):
    return [(i, i+1) for i in range(start, end)] + [(end, start)]

def make_mesh():
    points = [(-1, -1), (1, -1), (1, 1), (-1, 1)]
    facets = round_trip_connect(0, len(points)-1)
    facet_markers = [1, 2, 3, 4]

    circ_start = len(points)
    nsegments = 30
    points.extend(
            (0.25 * np.cos(angle), 0.25 * np.sin(angle))
            for angle in np.linspace(0, 2*np.pi, nsegments, endpoint=False))

    facets.extend(round_trip_connect(circ_start, len(points)-1))
    facet_markers.extend([-1] * nsegments)

    def needs_refinement(vertices, area):
        bary = np.sum(np.array(vertices), axis=0)/3
        max_area = 0.01 + la.norm(bary, np.inf)*0.01
        return bool(area > max_area)

    info = triangle.MeshInfo()
    info.set_points(points)
    info.set_facets(facets, facet_markers=facet_markers)

    built_mesh = triangle.build(info, refinement_func=needs_refinement)
    plex = fd_mesh._from_cell_list(
        2, np.array(built_mesh.elements), np.array(built_mesh.points), COMM_WORLD)

    import firedrake.cython.dmcommon as dmcommon

    v_start, v_end = plex.getDepthStratum(0)   # vertices
    for facet, fmarker in zip(built_mesh.facets, built_mesh.facet_markers):
        vertices = [fvert + v_start for fvert in facet]
        if fmarker > 0:  # interior facets are negative above
            join = plex.getJoin(vertices)
            plex.setLabelValue(dmcommon.FACE_SETS_LABEL, join[0], fmarker)

    return Mesh(plex)
    
mesh = make_mesh()

triplot(mesh)
plt.legend()
Out[3]:
<matplotlib.legend.Legend at 0x7f75964a6bf0>

Function Space¶

In [4]:
V = FunctionSpace(mesh, 'Lagrange', 1)

RHS and Coefficient¶

In [5]:
x = SpatialCoordinate(mesh)
f = conditional(le(sqrt(x[0]**2 + x[1]**2), 0.25), 25.0, 0.0)
kappa = conditional(le(sqrt(x[0]**2 + x[1]**2), 0.25), 25.0, 0.1)

Boundary Conditions¶

In [6]:
g = interpolate(20.0 * (1-x[0]*x[0]), V)
bcg = DirichletBC(V, g, [1])
bcz = DirichletBC(V, 0.0, [2,3,4])
bc = [bcg, bcz]

Symbolic Trial and Test Functions¶

In [7]:
u = TrialFunction(V)
v = TestFunction(V)

v
Out[7]:
Argument(WithGeometry(FunctionSpace(<firedrake.mesh.MeshTopology object at 0x7f75964a6a40>, FiniteElement('Lagrange', triangle, 1), name=None), Mesh(VectorElement(FiniteElement('Lagrange', triangle, 1), dim=2), 1)), 0, None)

Weak form¶

Pieces:

  • inner
  • grad
  • dx
In [8]:
a = kappa*inner(grad(u), grad(v))*dx
L = f*v*dx

Solve and Plot¶

In [9]:
U = Function(V)
solve(a == L, U, bc)
In [10]:
tricontourf(U)
Out[10]:
<matplotlib.tri.tricontour.TriContourSet at 0x7f758d810dc0>
In [11]:
fig = plt.figure(figsize=(10, 10))
axes = fig.add_subplot(111, projection='3d')
trisurf(U, axes=axes)
Out[11]:
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7f758d813940>
In [ ]: