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"""This demo program solves the incompressible Navier-Stokes equations
on an L-shaped domain using Chorin's splitting method."""
# Copyright (C) 2010-2011 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/>.
#
# Modified by Mikael Mortensen 2011
#
# First added: 2010-08-30
# Last changed: 2011-06-30
# Begin demo
import matplotlib.pyplot as plt
from dolfin import *
# Print log messages only from the root process in parallel
parameters["std_out_all_processes"] = False;
# Load mesh from file
mesh = Mesh("../lshape.xml.gz")
# Define function spaces (P2-P1)
V = VectorFunctionSpace(mesh, "Lagrange", 2)
Q = FunctionSpace(mesh, "Lagrange", 1)
# Define trial and test functions
u = TrialFunction(V)
p = TrialFunction(Q)
v = TestFunction(V)
q = TestFunction(Q)
# Set parameter values
dt = 0.01
T = 3
nu = 0.01
# Define time-dependent pressure boundary condition
p_in = Expression("sin(3.0*t)", t=0.0, degree=2)
# Define boundary conditions
noslip = DirichletBC(V, (0, 0),
"on_boundary && \
(x[0] < DOLFIN_EPS | x[1] < DOLFIN_EPS | \
(x[0] > 0.5 - DOLFIN_EPS && x[1] > 0.5 - DOLFIN_EPS))")
inflow = DirichletBC(Q, p_in, "x[1] > 1.0 - DOLFIN_EPS")
outflow = DirichletBC(Q, 0, "x[0] > 1.0 - DOLFIN_EPS")
bcu = [noslip]
bcp = [inflow, outflow]
# Create functions
u0 = Function(V)
u1 = Function(V)
p1 = Function(Q)
# Define coefficients
k = Constant(dt)
f = Constant((0, 0))
# Tentative velocity step
F1 = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx + \
nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)
# Pressure update
a2 = inner(grad(p), grad(q))*dx
L2 = -(1/k)*div(u1)*q*dx
# Velocity update
a3 = inner(u, v)*dx
L3 = inner(u1, v)*dx - k*inner(grad(p1), v)*dx
# Assemble matrices
A1 = assemble(a1)
A2 = assemble(a2)
A3 = assemble(a3)
# Use amg preconditioner if available
prec = "amg" if has_krylov_solver_preconditioner("amg") else "default"
# Use nonzero guesses - essential for CG with non-symmetric BC
parameters['krylov_solver']['nonzero_initial_guess'] = True
# Create files for storing solution
ufile = File("results/velocity.pvd")
pfile = File("results/pressure.pvd")
# Time-stepping
t = dt
while t < T + DOLFIN_EPS:
# Update pressure boundary condition
p_in.t = t
# Compute tentative velocity step
b1 = assemble(L1)
[bc.apply(A1, b1) for bc in bcu]
solve(A1, u1.vector(), b1, "bicgstab", "default")
# Pressure correction
b2 = assemble(L2)
[bc.apply(A2, b2) for bc in bcp]
[bc.apply(p1.vector()) for bc in bcp]
solve(A2, p1.vector(), b2, "bicgstab", prec)
# Velocity correction
b3 = assemble(L3)
[bc.apply(A3, b3) for bc in bcu]
solve(A3, u1.vector(), b3, "bicgstab", "default")
# Save to file
ufile << u1
pfile << p1
# Move to next time step
u0.assign(u1)
t += dt
# Plot solution
plt.figure()
plot(p1, title="Pressure")
plt.figure()
plot(u1, title="Velocity")
plt.show()
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