Ficks2nd-circle/Readme.md

```python
from fenics import *
from dolfin import *
import mshr
import matplotlib.pyplot as plt
import numpy as np
```


```python
T = 60*60*5 #final step
num_steps = 50
dt = T/num_steps #step size
```


```python
#2 Create mesh and define function space
domain = mshr.Circle(Point(0.,0.),1.0,60)
mesh = mshr.generate_mesh(domain, 25)
V = FunctionSpace(mesh, 'Lagrange', 1) #Lagrange are triangular elements
plot(mesh)
plt.show()
```


![png](output_2_0.png)



```python
#3 Defining boundary conditions (Dirichlet)
D = 1.4E-7 #cm^2/s
u_D = Constant(0.1)

def Dirichlet_boundary(x, on_boundary):
    return on_boundary 

bc = DirichletBC(V, Constant(1), Dirichlet_boundary)
```


```python
#Defining initial values and variational problem
u_n = interpolate(u_D,V)

u = TrialFunction(V)
v = TestFunction(V)
f = Constant(0)

F = u*v*dx+D*dt*dot(grad(u), grad(v))*dx-(u_n+dt*f)*v*dx
a, L = lhs(F), rhs(F)
```


```python
#Resolution on time steps
u = Function(V)
t = 0

vtkfile = File('sol/solution.pvd')
for n in range(num_steps):
    #update current time
    t += dt
    u_D.t = t
    #compute solution
    solve(a==L, u, bc)
    #vtkfile << (u,t)
    plot(u)
    plt.show()
    #update previous solution
    u_n.assign(u)
```


![png](output_5_0.png)


![png](output_5_48.png)