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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() |