Heat diffusion with a mesh that changes size

mesh
1

Hi everyone,

I’m pretty new to fenics. I’ve been using fenics for 1d heat diffusion from one material into another. However, I also want the entire mesh to shrink in size with time, but I have no idea how to approach this problem. I’m not getting any errors because I don’t even know where to start with this, but here is some example code of my static heat diffusion:

from fenics import *
import numpy as np

tf = 20*10**-9     # final time
num_steps = 60    # number of time steps (60)
dt = tf / num_steps # time step size
tol = DOLFIN_EPS
degree = 1


# Define Thermodynamic Parameters
init_temp = 500
peak_temp = 20000
k_0 = 100
k_1 = 50
rho = 8000 
c = 320

l = 2*10**-6 # Length of iron (m)
lw = 4*10**-6 # Length of window (m)
tot_l = l + lw
meshpoints = 300
my_mesh = IntervalMesh ( meshpoints,0,tot_l )
V = FunctionSpace(my_mesh, 'P', degree)
# Define boundary condition
u_D = Expression(str(peak_temp),degree=degree)

def boundary ( x ):
    value = x[0] < tol
    return value

bc = DirichletBC(V, u_D, boundary)


# Define initial value
u_0 = Expression(str(init_temp), degree=degree)
u_n = interpolate(u_0, V)
#u_n = project(u_D, V)


# Define variational problem
u = Function(V)
v = TestFunction(V)
tc = 'x[0] >= l+tol ? k_1 : k_0'
kappa = Expression(tc, degree=degree,
tol=tol, k_1=k_1, k_0=k_0, l=l)


# Solve
F = u*v*dx + kappa/(rho*c)*dt*dot(grad(u), grad(v))*dx - (u_n)*v*dx

# Time-stepping and plotting
inc = 0.01*10**-6
loc = l - inc
points = np.linspace(0,tot_l,meshpoints)
times = np.linspace(0,tf, num_steps)
time_line = []
num_loops = 1
t=0

for loop in range(num_loops):
    t=0
    u_n = interpolate(u_0, V)
    F = u*v*dx + kappa/(rho*c)*dt*dot(grad(u), grad(v))*dx - (u_n)*v*dx
    time_line = []
    for n in range(num_steps):

        # Update current time
        t += dt
        u_D.t = t
        kappa.t = t
        print(t)
        # Compute solution
        solve(F==0, u, bc)

        # Plot solution
        #u_line = [u(point) for point in points]
        time_line.append(u(loc))

        # Update previous solution
        u_n.assign(u)
2

You can use ALE.move, see; How to translate a 2D mesh by a vector? - #2 by kamensky

3

Thank you! Is ALE.move something I can implement independent of my heat equation? Or do I have to incorporate a new displacement term in my variational form?

4

The most mathematically correct thing would be to incorporate u in your variational form

5

I’ve been working on this problem a bit, but I still can’t quite grasp how I can incorporate the displacement into my variational form