Professor of Computational Mathematics at Chalmers University of Technology

Hans Petter Langtangen and I are working on a new, revised, and extended version of the FEniCS Tutorial. The plan is two volumes to cover both basic and advanced FEniCS programming. Here’s a sample from Chapter 3: A Gallery of PDE Solvers:

And here’s a bit of code to go along with the movie.

```"""
FEniCS tutorial demo program: Incompressible Navier-Stokes equations
for flow around a cylinder using the Incremental Pressure Correction
Scheme (IPCS).

rho (u' + u . grad(u)) - div(sigma) = f
div(u) = 0
"""

from __future__ import print_function
from fenics import *
from mshr import *
import numpy as np

T = 5.0            # final time
num_steps = 5000   # number of time steps
dt = T / num_steps # time step size
rho = 1.0          # density
nu = 0.001         # kinematic viscosity

# Create mesh
channel = Rectangle(Point(0, 0), Point(2.2, 0.41))
cylinder = Circle(Point(0.2, 0.2), 0.05)
geometry = channel - cylinder
mesh = generate_mesh(geometry, 64)

# Define function spaces
V = VectorFunctionSpace(mesh, 'P', 2)
Q = FunctionSpace(mesh, 'P', 1)

# Define boundaries
inflow   = 'near(x[0], 0)'
outflow  = 'near(x[0], 2.2)'
walls    = 'near(x[1], 0) || near(x[1], 0.41)'
cylinder = 'on_boundary && x[0]>0.1 && x[0]<0.3 && x[1]>0.1 && x[1]<0.3'

# Define inflow profile
inflow_profile = ('4.0*1.5*x[1]*(0.41 - x[1]) / pow(0.41, 2)', '0')

# Define boundary conditions
bcu_inflow   = DirichletBC(V, Expression(inflow_profile, degree=2), inflow)
bcu_walls    = DirichletBC(V, Constant((0, 0)), walls)
bcu_cylinder = DirichletBC(V, Constant((0, 0)), cylinder)
bcp_outflow  = DirichletBC(Q, Constant(0), outflow)
bcu = [bcu_inflow, bcu_walls, bcu_cylinder]
bcp = [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
u0 = Function(V)
u1 = Function(V)
p0 = Function(Q)
p1 = Function(Q)

# Define expressions used in variational forms
U   = 0.5*(u0 + u)
n   = FacetNormal(mesh)
f   = Constant((0, 0))
k   = Constant(dt)
rho = Constant(rho)
nu  = Constant(nu)

def epsilon(u):

# Define stress tensor
def sigma(u, p):

# Define variational problem for step 1
F1 = rho*dot((u - u0) / k, v)*dx + rho*dot(grad(u0)*u0, v)*dx \
+ inner(sigma(U, p0), epsilon(v))*dx \
+ dot(p0*n, v)*ds - dot(nu*grad(U).T*n, v)*ds \
- dot(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)

# Define variational problem for step 2

# Define variational problem for step 3
a3 = dot(u, v)*dx
L3 = dot(u1, v)*dx - k*dot(grad(p1 - p0), 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]

# Create VTK files for saving solution
vtkfile_u = File('velocity.pvd')
vtkfile_p = File('pressure.pvd')

# Create progress bar
progress = Progress('Time-stepping')
set_log_level(PROGRESS)

# Time-stepping
t = 0
for n in xrange(num_steps):

# Update current time
t += dt

# Step 1: Tentative velocity step
b1 = assemble(L1)
[bc.apply(b1) for bc in bcu]
solve(A1, u1.vector(), b1, 'bicgstab')

# Step 2: Pressure correction step
b2 = assemble(L2)
[bc.apply(b2) for bc in bcp]
solve(A2, p1.vector(), b2, 'bicgstab')

# Step 3: Velocity correction step
b3 = assemble(L3)
solve(A3, u1.vector(), b3, 'bicgstab')

# Plot solution
plot(u1, title='Velocity')
plot(p1, title='Pressure')

# Save solution to file
vtkfile_u << u1
vtkfile_p << p1

# Update previous solution
u0.assign(u1)
p0.assign(p1)

# Update progress
progress.update(t / T)

# Hold plot
interactive()
```

1. #### Chaitanya Goyal May 7, 2016

Hello Dr. Logg,
We are very grateful to you and your associates for making fenics open source with so many tutorials and demos. I am a MSc student trying to implement wave equation in elasoplastic media using fenics. I am finding it very difficult owing to the lack of detailed documentation and undocumented demos. I know the team stays very busy with so much work but it would be great if you could find some time for development and refinement of FSM app in future.
Thanks and Regards,

• #### loggMay 8, 2016 — Post Author

The new tutorial will contain many new examples. Maybe not exactly the one you are looking for… but hopefully it will help you figure out how to implement your particular problem in FEniCS.

2. #### Sophia May 18, 2016

Hello Dr. Logg,
What is the likely release date for the new tutorial?
Thank you,

• #### loggMay 20, 2016 — Post Author

I hope the first volume will be online within 1-2 weeks.

3. #### Jonas August 19, 2016

Hi Dr. Logg, I was wondering if there was an update on the release of the new tutorial. Thanks, Jonas

• #### loggAugust 22, 2016 — Post Author

Yes, we’re currently proofreading it. It should be ready within 1-2 weeks.