Hello,
I would like to convert the heat equation into a time dependent function. Please how would I do this ? I have found this code used in firedrakeproject for time dependent boundaries but not sure if it would serve any purpose in fenics.
c = Constant(sin(t))
bc = DirichletBC(V, 0, boundary)
while T_initial < T_final:
solve(F == 0, bcs=[bc])
T_initial += dt
c.assign(sin(T_initial))
As for my regular code, I used the fenics heat equation code and made a few edits which I have provided below.
Thanks,
FEniCS tutorial demo program: Diffusion of a Gaussian hill.
u'= Laplace(u) + f in a square domain
u = u_D on the boundary
u = u_0 at t = 0
u_D = f = 0
The initial condition u_0 is chosen as a Gaussian hill.
from dolfin import *
from mshr import *
from fenics import *
import time
# Define time things.
T_initial = 0.0 # initial time
T_final = 2 # final time
num_steps = 50 # number of time steps
dt = ( T_final - T_initial ) / num_steps # time step size
#Making a cylindrical geometry 10 cm radius and 15 cm height in S.I
cylinder = Cylinder(Point(0, 0, -7.5), Point(0, 0, 7.5), 5, 5)
domain = cylinder
# Making Mesh ( THe value corresponds to the mesh density)
mesh = generate_mesh(domain, 15) # generates 3D model of the cylindrical geometry in x, y, and z
axes.
V = FunctionSpace(mesh, 'P', 1)
# Define boundary condition
def boundary(x, on_boundary):
return on_boundary
# Boundary Conditions
bc = DirichletBC(V, Constant(0), boundary)
# Definining the intial temperature. 300 K
u_0 = Constant('300')
a = 5
# Expression('300', degree=2, a=5)
u_n = interpolate(u_0, V)
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(0)
# ( not sure what this really represents ) turn the F function as time dependent function
F = u*v*dx + dt*dot(grad(u), grad(v))*dx - (u_n + dt*f)*v*dx
a, L = lhs(F), rhs(F)
# Create VTK file for saving solution
vtkfile = File('heat_gaussian/solution.pvd')
# Time-stepping
u = Function(V)
T_inital = 0
for n in range(num_steps):
# Update current time
T_inital += dt
# Compute solution
solve(a == L, u, bc)
# Save to file and plot solution
vtkfile << (u, T_inital)
plot(u)
# Update previous solution
u_n.assign(u)
import matplotlib.pyplot as plt
plt.show()
‘’’
