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"""This demo solves the Stokes equations, using quadratic elements for
the velocity and first degree elements for the pressure (Taylor-Hood
elements). The sub domains for the different boundary conditions
used in this simulation are computed by the demo program in
src/demo/mesh/subdomains."""
# Copyright (C) 2007 Kristian B. Oelgaard
#
# 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/>.
#
# Modified by Anders Logg, 2008-2009.
#
# First added: 2007-11-16
# Last changed: 2009-11-26
# Begin demo
import matplotlib.pyplot as plt
from dolfin import *
# Load mesh and subdomains
mesh = Mesh("../dolfin_fine.xml.gz")
sub_domains = MeshFunction("size_t", mesh, "../dolfin_fine_subdomains.xml.gz")
plt.figure()
plot(mesh)
plt.figure()
plot(sub_domains)
# Define function spaces
P2 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
TH = P2 * P1
W = FunctionSpace(mesh, TH)
# No-slip boundary condition for velocity
# x1 = 0, x1 = 1 and around the dolphin
noslip = Constant((0, 0))
bc0 = DirichletBC(W.sub(0), noslip, sub_domains, 0)
# Inflow boundary condition for velocity
# x0 = 1
inflow = Expression(("-sin(x[1]*pi)", "0.0"), degree=2)
bc1 = DirichletBC(W.sub(0), inflow, sub_domains, 1)
# Collect boundary conditions
bcs = [bc0, bc1]
# Define variational problem
(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)
f = Constant((0, 0))
a = (inner(grad(u), grad(v)) - div(v)*p + q*div(u))*dx
L = inner(f, v)*dx
# Compute solution
w = Function(W)
solve(a == L, w, bcs)
# Split the mixed solution using deepcopy
# (needed for further computation on coefficient vector)
(u, p) = w.split(True)
print("Norm of velocity coefficient vector: %.15g" % u.vector().norm("l2"))
print("Norm of pressure coefficient vector: %.15g" % p.vector().norm("l2"))
# # Split the mixed solution using a shallow copy
(u, p) = w.split()
# Save solution in VTK format
ufile_pvd = File("velocity.pvd")
ufile_pvd << u
pfile_pvd = File("pressure.pvd")
pfile_pvd << p
# Plot solution
plt.figure()
plot(u, title="velocity")
plt.figure()
plot(p, title="pressure")
# Display plots
plt.show()
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