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# Copyright (C) 2011 Marie E. Rognes
#
# 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: 2011-11-09
# Last changed: 2012-08-06
# Begin demo
from dolfin import *
# Create classes for defining parts of the boundaries and the interior
# of the domain
class Left(SubDomain):
def inside(self, x, on_boundary):
return near(x[0], 0.0)
class Right(SubDomain):
def inside(self, x, on_boundary):
return near(x[0], 1.0)
class Bottom(SubDomain):
def inside(self, x, on_boundary):
return near(x[1], 0.0)
class Top(SubDomain):
def inside(self, x, on_boundary):
return near(x[1], 1.0)
class Obstacle(SubDomain):
def inside(self, x, on_boundary):
return (between(x[1], (0.5, 0.7)) and between(x[0], (0.2, 1.0)))
# Initialize sub-domain instances
left = Left()
top = Top()
right = Right()
bottom = Bottom()
obstacle = Obstacle()
# Define mesh
mesh = UnitSquareMesh(64, 64)
# Initialize mesh function for interior domains
domains = MeshFunction("size_t", mesh, mesh.topology().dim())
domains.set_all(0)
obstacle.mark(domains, 1)
# Initialize mesh function for boundary domains
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim()-1)
boundaries.set_all(0)
left.mark(boundaries, 1)
top.mark(boundaries, 2)
right.mark(boundaries, 3)
bottom.mark(boundaries, 4)
# Define input data
a0 = Constant(1.0)
a1 = Constant(0.01)
g_L = Expression("- 10*exp(- pow(x[1] - 0.5, 2))", degree=2)
g_R = Constant(1.0)
f = Constant(1.0)
# Define function space and basis functions
V = FunctionSpace(mesh, "CG", 2)
u = TrialFunction(V)
v = TestFunction(V)
# Define Dirichlet boundary conditions at top and bottom boundaries
bcs = [DirichletBC(V, 5.0, boundaries, 2),
DirichletBC(V, 0.0, boundaries, 4)]
# Define new measures associated with the interior domains and
# exterior boundaries
dx = Measure('dx', domain=mesh, subdomain_data=domains)
ds = Measure('ds', domain=mesh, subdomain_data=boundaries)
# Define variational form
F = (inner(a0*grad(u), grad(v))*dx(0) + inner(a1*grad(u), grad(v))*dx(1)
- g_L*v*ds(1) - g_R*v*ds(3)
- f*v*dx(0) - f*v*dx(1))
# Separate left and right hand sides of equation
a, L = lhs(F), rhs(F)
# Solve problem
u = Function(V)
solve(a == L, u, bcs)
# Evaluate integral of normal gradient over top boundary
n = FacetNormal(mesh)
m1 = dot(grad(u), n)*ds(2)
v1 = assemble(m1)
print("\int grad(u) * n ds(2) = ", v1)
# Evaluate integral of u over the obstacle
m2 = u*dx(1)
v2 = assemble(m2)
print("\int u dx(1) = ", v2)
# Plot solution
import matplotlib.pyplot as plt
plt.figure()
plot(u, title="Solution u")
# Plot solution and gradient
plt.figure()
plot(grad(u), title="Projected grad(u)")
# Show plots
plt.show()
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