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a516f1b Nov 25, 2016
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"""
FEniCS tutorial demo program: Incompressible Navier-Stokes equations
for channel flow (Poisseuille) on the unit square using the
Incremental Pressure Correction Scheme (IPCS).
u' + u . nabla(u)) - div(sigma(u, p)) = f
div(u) = 0
"""
from __future__ import print_function
from fenics import *
import numpy as np
T = 10.0 # final time
num_steps = 500 # number of time steps
dt = T / num_steps # time step size
mu = 1 # kinematic viscosity
rho = 1 # density
# Create mesh and define function spaces
mesh = UnitSquareMesh(16, 16)
V = VectorFunctionSpace(mesh, 'P', 2)
Q = FunctionSpace(mesh, 'P', 1)
# Define boundaries
inflow = 'near(x[0], 0)'
outflow = 'near(x[0], 1)'
walls = 'near(x[1], 0) || near(x[1], 1)'
# Define boundary conditions
bcu_noslip = DirichletBC(V, Constant((0, 0)), walls)
bcp_inflow = DirichletBC(Q, Constant(8), inflow)
bcp_outflow = DirichletBC(Q, Constant(0), outflow)
bcu = [bcu_noslip]
bcp = [bcp_inflow, bcp_outflow]
# Define trial and test functions
u = TrialFunction(V)
v = TestFunction(V)
p = TrialFunction(Q)
q = TestFunction(Q)
# Define functions for solutions at previous and current time steps
u_n = Function(V)
u_ = Function(V)
p_n = Function(Q)
p_ = Function(Q)
# Define expressions used in variational forms
U = 0.5*(u_n + u)
n = FacetNormal(mesh)
f = Constant((0, 0))
k = Constant(dt)
mu = Constant(mu)
rho = Constant(rho)
# Define strain-rate tensor
def epsilon(u):
return sym(nabla_grad(u))
# Define stress tensor
def sigma(u, p):
return 2*mu*epsilon(u) - p*Identity(len(u))
# Define variational problem for step 1
F1 = rho*dot((u - u_n) / k, v)*dx + \
rho*dot(dot(u_n, nabla_grad(u_n)), v)*dx \
+ inner(sigma(U, p_n), epsilon(v))*dx \
+ dot(p_n*n, v)*ds - dot(mu*nabla_grad(U)*n, v)*ds \
- dot(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)
# Define variational problem for step 2
a2 = dot(nabla_grad(p), nabla_grad(q))*dx
L2 = dot(nabla_grad(p_n), nabla_grad(q))*dx - (1/k)*div(u_)*q*dx
# Define variational problem for step 3
a3 = dot(u, v)*dx
L3 = dot(u_, v)*dx - k*dot(nabla_grad(p_ - p_n), v)*dx
# Assemble matrices
A1 = assemble(a1)
A2 = assemble(a2)
A3 = assemble(a3)
# Apply boundary conditions to matrices
[bc.apply(A1) for bc in bcu]
[bc.apply(A2) for bc in bcp]
# Time-stepping
t = 0
for n in range(num_steps):
# Update current time
t += dt
# Step 1: Tentative velocity step
b1 = assemble(L1)
[bc.apply(b1) for bc in bcu]
solve(A1, u_.vector(), b1)
# Step 2: Pressure correction step
b2 = assemble(L2)
[bc.apply(b2) for bc in bcp]
solve(A2, p_.vector(), b2)
# Step 3: Velocity correction step
b3 = assemble(L3)
solve(A3, u_.vector(), b3)
# Plot solution
plot(u_)
# Compute error
u_e = Expression(('4*x[1]*(1.0 - x[1])', '0'), degree=2)
u_e = interpolate(u_e, V)
error = np.abs(u_e.vector().array() - u_.vector().array()).max()
print('t = %.2f: error = %.3g' % (t, error))
print('max u:', u_.vector().array().max())
# Update previous solution
u_n.assign(u_)
p_n.assign(p_)
# Hold plot
interactive()