Copyright (C) 2020 Andreas Kloeckner
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.
from firedrake import *
import numpy as np
import numpy.linalg as la
import firedrake.mesh as fd_mesh
import matplotlib.pyplot as plt
from firedrake.__future__ import interpolate
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.plex_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.gca().set_aspect("equal")
plt.legend()
<matplotlib.legend.Legend at 0x7f7439f019d0>
V = FunctionSpace(mesh, 'Lagrange', 1)
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)
g = assemble(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]
#clear
u = TrialFunction(V)
v = TestFunction(V)
v
Argument(WithGeometry(FunctionSpace(<firedrake.mesh.MeshTopology object at 0x7f738079dfd0>, FiniteElement('Lagrange', triangle, 1), name=None), Mesh(VectorElement(FiniteElement('Lagrange', triangle, 1), dim=2), 9)), 0, None)
#clear
a = kappa*inner(grad(u), grad(v))*dx
L = f*v*dx
U = Function(V)
solve(a == L, U, bc)
tricontourf(U)
<matplotlib.tri._tricontour.TriContourSet at 0x7f7375169e10>
fig = plt.figure(figsize=(10, 10))
axes = fig.add_subplot(111, projection='3d')
trisurf(U, axes=axes)
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7f7374640f10>