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# Copyright (C) 2015 Anders Logg
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see <http://www.gnu.org/licenses/>.
#
# First added: 2015-11-05
# Last changed: 2015-11-17
#
# This demo program solves Poisson's equation on a domain defined by
# three overlapping and non-matching meshes. The solution is computed
# on a sequence of rotating meshes to test the multimesh
# functionality.
from dolfin import *
class DirichletBoundary(SubDomain):
def inside(self, x, on_boundary):
return on_boundary
def solve_poisson(t, x1, y1, x2, y2):
"Compute solution for given mesh configuration"
# Create meshes
r = 0.5
mesh_0 = RectangleMesh(Point(-r, -r), Point(r, r), 16, 16)
mesh_1 = RectangleMesh(Point(x1 - r, y1 - r), Point(x1 + r, y1 + r), 8, 8)
mesh_2 = RectangleMesh(Point(x2 - r, y2 - r), Point(x2 + r, y2 + r), 8, 8)
mesh_1.rotate(70*t)
mesh_2.rotate(-70*t)
# Build multimesh
multimesh = MultiMesh()
multimesh.add(mesh_0)
multimesh.add(mesh_1)
multimesh.add(mesh_2)
multimesh.build()
# Create function space
V = MultiMeshFunctionSpace(multimesh, "Lagrange", 1)
# Define trial and test functions and right-hand side
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(1)
# Define facet normal and mesh size
n = FacetNormal(multimesh)
h = 2.0*Circumradius(multimesh)
h = (h('+') + h('-')) / 2
# Set parameters
alpha = 10.0
beta = 1.0
# Define bilinear form
a = dot(grad(u), grad(v))*dX \
- dot(avg(grad(u)), jump(v, n))*dI \
- dot(avg(grad(v)), jump(u, n))*dI \
+ alpha/h * jump(u)*jump(v)*dI \
+ beta/h**2 * dot(jump(u), jump(v))*dO
# Define linear form
L = f*v*dX
# Assemble linear system
A = assemble_multimesh(a)
b = assemble_multimesh(L)
# Apply boundary condition
zero = Constant(0)
boundary = DirichletBoundary()
bc = MultiMeshDirichletBC(V, zero, boundary)
bc.apply(A, b)
# Lock inactive dofs
V.lock_inactive_dofs(A, b)
# Compute solution
u = MultiMeshFunction(V)
solve(A, u.vector(), b)
return u
if MPI.size(MPI.comm_world) > 1:
info("Sorry, this demo does not (yet) run in parallel.")
exit(0)
# Parameters
T = 10.0
N = 100
dt = T / N
# Create files for output
f0 = XDMFFile("output/u0.xdmf")
f1 = XDMFFile("output/u1.xdmf")
f2 = XDMFFile("output/u2.xdmf")
# Iterate over configurations
for n in range(N):
info("Computing solution, step %d / %d." % (n + 1, N))
# Compute coordinates for meshes
t = dt*n
x1 = sin(t)*cos(2*t)
y1 = cos(t)*cos(2*t)
x2 = cos(t)*cos(2*t)
y2 = sin(t)*cos(2*t)
# Compute solution
u = solve_poisson(t, x1, y1, x2, y2)
# Save to file
f0.write(u.part(0, deepcopy=True), t)
f1.write(u.part(1, deepcopy=True), t)
f2.write(u.part(2, deepcopy=True), t)
# Close files
f0.close()
f1.close()
f2.close()
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