Hi everyone,

I am currently trying to model the cooling of a cylinder via radiation as a function of time using the heat equation, where the emissivity of the material `epsilon_eff`

is dependent on the temperature. However, `epsilon_eff`

seems not to be updating for each time step. The basic code is below:

```
from fenics import *
from dolfin import *
# Define the domain
r1 = 1.0 # inner radius
r2 = 2.0 # outer radius
L = 3.0 # length
N = 20 # number of mesh elements
# Create the mesh
mesh = CylinderMesh(Point(0, -L/2, 0), Point(0, L/2, 0), r1, r2, N)
# Define the function space
V = FunctionSpace(mesh, 'P', 1)
# emissivity function
emiss_in = [0.03,0.042,0.054,0.062,0.080,0.091,0.109,0.116,0.135,0.194,0.213,0.232,0.251]
temp_emiss = [300,400,500,600,700,800,900,1000,1100,1600,1800,2000,2200]
emiss_f = UnivariateSpline(temp_emiss, emiss_in, k = 2, ext = 3)
# Define the initial temperature
u0 = Expression('1000', degree=1)
# Define the boundary conditions
bc = DirichletBC(V, u0, 'on_boundary')
# Define the thermal conductivity and radiation coefficient
k = Constant(1.0)
sigma = Constant(5.67e-8)
# Define the variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(0.0)
emiss_eff_array = emiss_f(u.vector().get_local())
emiss_space_eff = FunctionSpace(mesh, 'CG',1)
epsilon_eff = Function(emiss_space_eff)
epsilon_eff.vector()[:] = emiss_eff_array
a = k * inner(grad(u), grad(v)) * dx
L = f * v * dx - epsilon_eff * sigma * u**4 * v * ds
# Define the time-stepping method
T = 100.0 # total simulation time
dt = 0.1 # time step
t = 0.0 # initial time
u = Function(V)
# Create the file for storing the solution
file = File('cooling.pvd')
# Time-stepping loop
while t < T:
print(t)
t += dt
solve(a == L, u, bc)
print(max(u.vector()))
print(max(epsilon_eff.vector()))
file << (u, t)
```

I think that the issue maybe with the line, `epsilon_eff.vector()[:] = emiss_eff_array`

but I am unsure about how this can be fixed.

Thanks in advance